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A  V THOR 


JOHNSON,  WILLIAM 
ERNEST 


TITLE: 


LOGIC... 


PLACE: 


CAMBRIDGE 


DA  TE : 


1921-24 


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Johnson,  William  ErnesI,  1859-1931 , 

Logic  .„,  by  W.  E.  Johnson  ...     Carnbrid-e.  The  rniver^itv 

Cc|:y  in  Butler,      1921-24.     3  t. 


_1.  Logic. 


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LOGIC 


PART  I 


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LOGIC 


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2- 


CAMBRIDGE  UNIVERSITY  PRESS 

C.  F.  CLAY,  Manager 

LONDON   :  FETTER  LANE,  E.C.  4 


NEW  YORK  :  THE  MACM'ILLAN  CO. 

BOMBAY 

CALCUTTA     MACMILLAN  AND  CO.,  Ltd. 

MADRAS 

TORONTO    :  THE  MACMILLAN  CO.  OF 

CANADA,  Ltd. 
TOKYO:  MARUZEN-KABUSHIKI-KAISHA 


PART  I 


BY 


W.  E.  JOHNSON,  M.A. 

FELLOW  OF  king's  COLLEGE,  CAMBRIDGE, 

SIDGWICK  LECTURER  IN  MORAL  SCIENCE  IN  THE 
UNIVERSITY  OF  CAMBRIDGE. 


n^NTA  p€T  €l  M>^  t6  AAHGtC 


ALL  RIGHTS  RESERVED 


CAMBRIDGE 
AT  THE  UNIVERSITY  PRESS 

1921 


w.  / 


'•  MAN  »4 ''rational  ANIMAL" 


M 


PREFACE 

THE  present  work  is  intended  to  cover  the  whole 
field  of  Logic  as  ordinarily  understood.  It  includes 
an  outline  of  elementary  Formal  Logic,  which  should 
be  read  in  close  connection  with  Dr  Keynes's  classical 
work,  in  which  the  last  word  has  been  said  on  most 
of  the  fundamental  problems  of  the  subject.  As  regards 
Material  Logic,  I  have  taken  Mill's  System  of  Logic 
as  the  first  basis  of  discussion,  which  however  is  sub- 
jected to  important  criticisms  mostly  on  the  lines  of 
the  so-called  conceptualist  logicians. 

I  have  to  express  my  great  obligations  to  my  former 
pupil.  Miss  Naomi  Bentwich,  without  whose  encourage- 
ment and  valuable  assistance  in  the  composition  and 
arrangement  of  the  work,  it  would  not  have  been  pro- 
duced in  its  present  form. 

W.  E.  J. 

March  30,  192 1. 


Pa-f-^.     I   .  "  F»Y^»-<tl    Hr%4.c/- 


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CONTENTS 


INTRODUCTION 

PAGE 

1.  Definition  of  Logic.    Grounds  for  including  the  theory  of  inductitm       .     xiii 

2.  Thinking  includes  perceptual  judgments xvii 

3.  All  ulterior  motives  of  the  pursuit  of  truth  irrelevant  to  the  analysis  and 
criticism  of  thought xvii 

4.  Relation  of  Logic  to  the  Art  of  Thinking xix 

5.  Logic,  Aesthetics  and  Ethics  as  the  three  normative  studies  .        .        .  xx 

6.  Relation  of  Logic  to  Universal  Grammar xxi 

7.  Logic  and  Mathematics.   Methodology.   Pure  and  Applied  Mathematics  xxii 

8.  Historical  sketch  of  the  problems  connecting  Logic  with  Philosophy  and 
with  Psychology.  Realism,  Conceptualism  and  Nominalism.  Material- 
istic and  Empirical  Logic xxvii 

9.  Special  features  of  the  author's  treatment  of  Logic        .        .        .  xxxiii 


CHAPTER  I 

THE  PROPOSITION 

1.  Sentence,  assertion  and  proposition i 

2.  Assertion  as  conscious  belief 4 

3.  Necessity  for  recognising  the  mental  attitude  in  logic    ....  6 

4.  Relation  between  grammatical  and  logical  analysis       ....  8 

5.  Connected  functioning  of  substantive  and  adjective  in  the  proposition    .  9 

6.  Criticism  of  the  view  that  the  essential  nature  of  the  proposition  is  a 
statement  of  identity 13 

7.  The  proposition  r^arded  as  characterising  the  fact,  and  its  analogies 
with  the  adjective 14 


CHAPTER  ri 

THE  PRIMITIVE  PROPOSITION 

|§  I.  Account  of  the  psychologically  primitive  form  of  assertion — exclamatory 

or  impersonal 18 

If  2.  The  need  of  separate  presentment  for  the  most  elementary  forms  of 

judgment 10 

3.  CritietMD  o{  Mr  Bradley's  dictum  *  distinction  impli^  differeace '  .         .  4 a 

4.  Illustrations  of  less  primitive  forms  of  assertion 33 


y 


viii  CONTENTS 

V  CHAPTER  III  V 

COMPOUND  PROPOSITIONS 

PAGE 

§  I.  Definition  of  compound  and  simple  as  applied  to  propositions        .         .       26 

§  2.  Distinction  between  the  different  forms  of  compound  propositions ;  the 
logical  conjunctions;  distinction  between  the  enumerative  and  conjunctive 

*and' ^7 

§  3.  The  first  law  of  thought  called  *The  Law  of  Double  Negation'      .         .       29 

§4.  The  laws  of  conjunctive  propositions 29 

§  5.  The  four  forms  of  composite  proposition  and  their  immediate  implications  30 
§  6.  Complementary  and  supplementary  propositions  and  their  rules  .  .  35 
§  7.  Criticism  of  the  paradoxical  forms  of  the  composite,  and  preliminary 

explanation  of  the  solution  of  the  paradox 38 

§  8.  The  bearing  of  the  distinction  between  hypothesis  and  assertion  upon 

the  paradox 44 

§  9.  Interpretation  of  compound  propositions  as  expressing  possible  conjunc- 
tives and  necessary  composites.  Table  of  all  the  possible  relations  of 
one  proposition  to  another 47 

CHAPTER  IV 

SECONDARY  PROPOSITIONS  AND  MODALITY 

§  I.  Definition  of  primary  and  secondary  propositions  and  of  pre-propositional 

adjectives    ......••••••  5° 

§2.  Modaladjectivesasa  species  of  pre-propositional.   The  adjectives  'true' 

and 'false' in  this  connection 5^ 

§  3.  The  opposition  of  modal  adjectives 53 

§4.  Antithesis  between 'certified' and 'uncertified' 55 

§  5.  The  rule  that  holds  universally  between  two  antithetical  modals   .         .  56 

§6.  Threefold  meanings  of 'necessary' and 'possible'  .  •  •  •  59 
§7.  Antithesis  of  'certified'  and  'uncertified'  as  epistemic;    antithesis  of 

'  nomic '  and  '  contingent '  as  constitutive 61 

§  8.  Comparison  of  Mill's  distinction  between  verbal  and  real  propositions, 

with  Kant's  distinction  between  analytic  and  synthetic  propositions        .       62 

§  9.  Summary  of  the  treatment  of  modal  adjectives  in  connection  with  secon- 
dary propositions 65 

CHAPTER  V 

NEGATION 
§1.  The  nature  of  pure  negation  and  its  different  degrees  of  significance       .       ¥^ 

§  2.  The  importance  of  negation  determined  by  its  relevance  to  a  specific 

purpose 1  "9 

§  3.  Can  there  be  a  proposition  where  there  is  nothing  corresponding  to  the 
subject-term  ?  With  illustrations  of  elementary  forms  of  obversion  and 
contradiction y<5 

§  4.  An  account  of  the  incomplete  proposition  '  ^  is '  .         .         .         •  )) 

§5.  The  hypothetical  element  in  the  proposition  ' 6"  is '       .         .         •         •  7.\7 
§  6.  Summary :  determinateness  of  fact  contrasted  with  indeterminateness  of 

knowledge 76* 


CONTENTS 


IX 


CHAPTER  VI 
THE  PROPER  NAME  AND  THE  ARTICLE 

FACE 

§  I.  Distinction  between  the  proper  and  the  descriptive  name  ...  80 
§  2.  Proper  name  explained  in  terms  of  the  introductory  and  referential 

articles °^ 

§  3.  Distinction  between  the  connotation  and  the  etymology  of  a  name         .  84 

§  4.  Distinctions  between  the  four  kinds  of  article  or  applicatives  .        .  85 

§  5.  The  demonstrative  applicatives ^ 

§  6.  Ostensive  definition ^9 

CHAPTER  VII 
GENERAL  NAMES,  DEFINITION  AND  ANALYSIS 

§  I .  Connection  between  general  names  and  the  applicatives  •  •  •  97 
§  2.  Distinction  between  connotation  and  comprehension.   Can  adjectives 

be  predicated  of  abstract  names  ? 100 

§  3.  The  nature  of  the  analysis  involved  in  definition.   Definition  by  means 

of  substitution  of  phrase.   The  indefinable 103 

§  4.  Analytic  and  synthetic  modes  of  definition 106 

§  5.  Partition,  resolution  and  analysis 109 

CHAPTER  VIII 

ENUMERATIONS  AND  CLASSES 

§  I.  The  relation  'comprising'  as  defining  the  nature  of  an  enumeration  and 

an  item.   The  three  operators  for  enumerations 113 

§  2.  Enumerations  of  different  orders .116 

§  3.  Distinction  between  '  comprised  '  and  '  included '  .         .        .        .110 

§  4.  Connection  between  enumerations  and  classes  and  the  question  whether 

'  class '  is  a  genuine  entity 1*1 

§  5.  The  nature  and  reality  of  the  class  as  determined  by  the  type  of  adjective 

significantly  predicable  of  it '^6 

§6.  Attempted  proof  of  the  genuineness  of  'class'  by  the  so-called  principle 
of  Abstraction.  Charge  of  petitio  principii  and  ignoratio  elenchi  against 
the  alleged  proof '^^ 

V  CHAPTER  IX  X 

THE  GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS 

|§  1.  Pure  general  propositions.  Thing 130 

|§  2.  Analysis  of  the  general  proposition,  showing  how  adjectives  alone  fiinction 

in  the  predicate '3® 

3.  Different  interpretations  of  the  universal  and  particular  proposition,  and 

the  resultant  modifications  in  their  relations  of  implication  and  opposition     1 35 

J.  L.  ^ 


CONTENTS 


CONTENTS 


XI 


§  4.  Comparison  with  traditional  scheme 

§  5.  Development  of  the  immediate  implications  of  the  universal  proposition 

as  ordinarily  interpreted  ;  with  tables  of  immediate  inferences 
^  6.  Development  of  the  analogies  between  universals  and  particulars  on  the 

one  side  and  necessary  composites  and  possible  conjunctions  on  the  other 
%  7.  Formulation  of  general  propositions  in  terms  of  classes.    The  principle 

underlying  Euler's  and  Venn's  diagrams 

^  8.  Summary 


PAGE 

139 

140 

143 

146 

155 


CHAPTER  X 

EXISTENTIAL,  SUBSISTENTIAL  AND 
NARRATIVE  PROPOSITIONS 

§  I.  The  proper  principle  for  the  Classification  of  Propositions  .  .  •  156 
§2.  Philosophical  distinction  between  Existential  and  Subsistential  .  .158 
§  3.  The  so-called  existential  formulation  of  Propositions  :  Two  meanings  of 

the  Universe  of  Discourse ^59 

§4.  The  interpretation  of  mythical  propositions  as  elliptically  secondary       .     164 

§  5.  The  Narrative  Proposition '^6 

§  6.  Distinction  between  fictitious  and  historical  narratives  .  .  .  .167 
§  7.  The  relation  between  the  existence  of  a  class  and  the  existence  of  an 

individual '7^ 


u^CHAPTER  XI 

THE  DETERMINABLE 

§  I.  The  fundamentum  divisionis,  for  which  the  name  Determinable  is  to 
stand.  The  analogy  and  distinction  between  the  relation  of  an  individual 
to  its  class  and  that  of  an  adjectival  determinate  to  its  determinable      .     173 

§  2.  Can  abstracts  be  divided  into  singular  and  general?  The  corresponding 
distinction  is  properly  between  the  comparatively  determinate  and  the 
comparatively  indeterminate.  The  notion  of  a  determinable  as  generating 
its  determinates '77 

§  3.  The  increase  of  intension  that  determines  a  decrease  of  extension  to  be 
supplemented  by  the  superdeterminateness  of  adjectival  characterisation. 
Classification  as  starting  from  a  summum  genus  constituted  by  mere 
determinables  and  terminating  in  an  ultima  species^  constituted  by  abso- 
lute determinates.    Illustration  from  Botany  .         .         .         .         .178 

§  4.  Relations  between  determinates  under  a  given  determinable.  Incom- 
patibility. Order  of  qualitative  betweenness,  continuous  or  discrete. 
Complex  determinables 181 

§  5.  Important  consequences  of  the  distinction  between  the  absolutely  deter- 
minate and  the  comparatively  indeterminate  183 


CHAPTER  Xn 
THE  RELATION  OF  IDENTITY 

PAGE 

§  I.  Identity  goes  along  with  Otherness,  the  two  being  co-opponent  relations. 
Verbal  and  Factual  Identification  involve  the  same  kind  of  identity,  the 
nature  of  the  propositions  only  differing        .         .         .         .         .         .186 

§  1.  False  sense  in  which  identity  has  been  said  to  imply  difference      .         .     188 

§  3.  Relations  like  and  unlike ;  similar  and  dissimilar ;  agreeing  and  dis- 
agreeing  189 

§  4.  Adjectives  comparable  and  disparate.  Distensive  magnitude.   Difference 

properly  adjectival       .        ^ 190 

§  5.  Adjectival  comparison  implies  substantival  otherness.    False  view  of  the 

notion  of 'numerical' difference   ........     192 

§6.  Criticism  of  the  Leibnizian  dictum :  the  identity  of  indiscernibles  .     193 

§  7.  Can  identity  be  defined?   Identity  carries  co-implication  and  substituta- 

bility.    Exception  for  secondary  proposition  ......     195 

§  8.   Identity  and  the  Continuant,  psychical  or  physical.  Relation  of  Occurrent 

to  Continuant.  The  causal  bond  that  constitutes  the  unity  of  Experience     199 


CHAPTER  XIII 

RELATIONS  OR  TRANSITIVE  ADJECTIVES 

§  I.  Monadic,  diadic,  triadic, etc  adjectives  and  propositions.   Every  relative 

implies  its  correlative .     203 

§  2.  The  substantive-couple  characterised  by  an  adjective-couple  .         .         .     205 

§3.  Illustrations  of 'analogy' extending  beyond  arithmetical  ratio        .         .     207 

§  4.  The  coupling-tie  and  the  characterising-tie,  expressed  in  the  grammatical 

rules  of  governance  and  accordance      . 209 

§  5.  The  paradox  of  relation.  Distinction  between  tie  and  relation.  The 
relation  'characterised'  as  unit  relation  entering  into  all  adjectives  and 
relations 211 

§  6.   How  adjectives,  relations  and  propositions  may  function  as  substantives     214 

§  7.  Relational  propositions  of  triadic  and  higher  orders.  The  cognates  of  a 
relation :  permutations  and  bracketing.  Relations  between  terms  and 
possibilia     .         .         .         .         .         .         .         .         .         .         .         .216 

§  8.  The  problem  of  the  relation  of  assertion  to  the  proposition.    How  ideas 

are  involved  in  propositions 219 


CHAPTER  XIV 

LAWS  OF  THOUGHT 

§  1.  The  logical  meaning  of  consistency  as  distingfuished  from  truth.    The 

Laws  of  Thought  may  possibly  be  not  enumerable  as  independents        .     222 

§2.  The  value  and  functions  of  truisms 223 

§  3.  In  what  sense  are  the  principles  of  logic  imperatives  ?  The  distinction 
between  normative  and  positive  sciences  dealing  with  psychological 
material 224 

^2 


xii  CONTENTS 

PAGE 

§  4.  The  fundamental  formulae  as  used  implicitiy  in  building  up  the  logical 

system  itself        ............     "226 

§5.  The  three  laws  of  identity :  Transitive,  Symmetrical,  Reflexive     .         .     217 

§  6.  The  four  Principles  of  Propositional  Determination  ;  and  their  import. 
Expression  of  the  same  for  Adjectival  Determination.  These  in 
generalised  form.  The  distinction  between  false  and  not-true,  true  and 
not-false.   These  compared  with  male  and  female  ....     228 

§  7.  Defence  of  the  first  Principle  of  Propositional  Determination.    Truth  as 

temporally  unalterable  and  as  capable  of  only  one  meaning   .         .         •     ^33 

§  8.  The  four  Principles  of  Adjectival  Determination,  as  bringing  out  the 
relation  of  an  adjective  to  its  determinable.  Postulate  of  the  absolutely 
determinate  character  of  the  real.  Substantival  Categories  distinguished 
by  the  Adjectival  Determinables  by  which  they  are  characteri sable.  The 
possibility  of  exhaustively  comprehending  the  range  of  variation  of  a 
determinable.    Summary       .         .         .         .         .         .         .         .         >     "^37 

§  9.  The  four  Principles  of  Connectional  Determination.  These  embody  the 
purely  formal  properties  of  the  causal  relation.  Its  complementary  aspects 
— Agreement  and  Difference.  The  fundamental  postulate  involved  in 
the  existence  of  Laws  of  Nature.  The  plurality  and  uniqueness  both  of 
cause  and  of  effect       ..........     242 

§  10.  The  bearing  of  these  principles  upon  the  problem  of  internal  and  external 

relations 249 

INDEX .     253 


INTRODUCTION 

§  I .  Logic  is  most  comprehensively  and  least  contro- 
versially defined  as  the  analysis  and  criticism  of  thought. 
This  definition  involves  the  least  possible  departure 
from  the  common  understanding  of  the  term  and  is  not 
intended  to  restrict  or  extend  its  scope  in  any  unusual 
way.  The  scope  of  logic  has  tended  to  expand  in  two 
directions — backwards  into  the  domain  of  metaphysics, 
and  forwards  into  that  of  science.  These  tendencies 
show  that  no  rigid  distinction  need  be  drawn  on  the 
one  side  between  logic  and  metaphysics,  nor  on  the 
other  between  logic  and  science.  The  limits  imposed 
by  any  writer  are  justified  so  far  as  his  exposition  ex- 
hibits unity  ;  it  is,  in  fact,  much  more  important  to 
remove  confusions  and  errors  within  the  subjects  dis- 
cussed under  the  head  of  logic,  than  to  assign  precise 
limits  to  its  scope.  It  is,  I  hold,  of  less  importance  to 
determine  the  line  of  demarcation  between  logic  and 
philosophy  than  that  between  logic  and  science ;  so 
that  my  treatment  of  logic  might  be  called  philosophical 
in  comparison  with  that  of  those  who  implicitly  or  ex- 
plicitly separate  their  criticism  and  analysis  from  what 
in  the?r  view  should  be  relegated  to  epistemology  and 
ontology. 

This  account  of  the  scope  of  logic  does  not  differ  in 
any  essential  respects  from  that  given,  for  example,  in 
Mill's  long  introductory  chapter.  The  special  feature  of 
Mill's  logic  is  the  great  prominence  given  to  the  theory 
of  induction,  in  contrast  to  most  of  his  predecessors 


XIV 


INTRODUCTION 


INTRODUCTION 


XV 


and  contemporaries,  including  Whately.  Whately  does 
not  omit  reference  to  induction  any  more  than  Mill 
omits  syllogism  :  where  they  differ  is  that  Whately 
asserts  that  in  order  to  be  valid  any  inductive  inference 
must  be  formulated  syllogistically,  and  that  therefore 
the  principle  for  induction  is  dependent  on  the  principle 
of  syllogism.  Mill  opposes  this  view  ;  but  as  regards 
the  scope  of  logic  there  is  no  disagreement  between 
them  :  they  differ  simply  on  the  question  of  the  rela- 
tions of  deduction  to  induction. 

If  any  writer  deliberately  or  on  principle  dismisses 
from  logic  the  theory  of  inductive  inference,  it  must  be 
on  one  of  three  grounds  :  either  (a)  that  no  inductive 
inference  is  valid  ;  or  (d)  that  different  criteria  of  validity 
apply  to  different  sciences  ;  or  (c)  that  the  problem  of 
the  validity  of  induction  constitutes  a  topic  to  be  in- 
cluded in  some  study  other  than  that  named  logic.  As 
regards  (a),  this  is  the  view  which  seems  to  be  held  by 
Venn  in  his  Empirical  Logic  where,  in  the  chapter  on 
the  subjective  foundations  of  induction,  he  acknow- 
ledges that  as  a  matter  of  fact  human  beings  do  make 
directly  inductive  inferences,  even  with  a  feeling  of 
conviction,  but  that  no  warrant  for  such  conviction  can 
be  found.  Another  aspect  of  his  view  of  induction  is 
expounded  in  the  chapter  on  the  objective  foundations 
of  induction,  in  which  he  classifies  the  different  kinds 
of  uniformity — such  as  sequence,  co-existence,  perma- 
nence, rhythm — which  are  used  as  major  premisses, 
expressive  of  actual  fact,  by  means  of  which  specific 
uniformities  under  each  general  head  are  established  as 
valid.  When  then  he  is  asked  what  reasonable  ground 
there  is  for  accepting  these  major  premisses  as  true,  he 


maintains  in  effect  that  they  have  to  be  assumed,  in 
order  to  give  security  to  the  conclusions  inductively 
inferred.  In  using  the  word  assumption,  there  seems 
to  be  some  ambiguity,  namely  whether  it  is  to  be  under- 
stood to  mean  *  assumed  to  be  true  although  known 
to  be  false '  or  '  assumed  to  be  true  although  unprov- 
able.' I  take  Venn  to  mean  the  latter,  and  that  the 
attitude  towards  this  assumption  is  merely  one  of  felt 
certainty — felt,  indeed,  by  all  human  beings,  but  having 
no  root  in  our  rational  nature,  and  only  exhibiting  a 
common  psychological  disposition  or  character.  This 
view,  that  there  is  no  inductive  principle  that  is  self- 
evidently  or  demonstrably  true,  seems  to  be  held  by 
many  other  logicians,  though  none  of  them,  I  think, 
put  it  as  explicitly  as  Venn.  So  while  he  and  others 
include  induction  in  their  logical  exposition,  they  neg- 
lect what  I  take  to  be  the  essential  justification  for 
its  inclusion,  namely  as  affording  a  systematic  criticism 
of  the  question  of  its  validity.  As  regards  (^),  many 
excellent  text-books  have  been  written  in  these  days 
treating  of  the  principles  and  methods  peculiar  to 
different  sciences  ;  it  is  not  denied  by  their  authors 
that  this  treatment  is  logical ;  but,  if  not  explicitly 
stated,  yet  it  seems  to  be  suggested  that  in  comparing 
the  logic  of  one  science  with  that  of  another  the  sole 
result  (s  to  exhibit  differences,  and  that  no  one  set  of 
principles  applies  to  all  the  different  sciences.  If  this 
were  the  fact  there  would  be  some  excuse  for  excluding 
the  treatment  of  induction  from  the  scope  of  logic,  on 
the  ground  that  the  discussion  of  each  of  the  separate 
principles  should  be  relegated  to  its  own  department 
of  science.    But  if,  as  I  hold  in  agreement  with  most 


lA.    fffinr»40 


XVI 


INTRODUCTION 


INTRODUCTION 


xvn 


other  logicians,  there  must  be  a  community  of  principle 
discoverable  in  all  sciences,  then  the  discussion  of  this 
must  be  included  in  logic.  As  regards  (c)  the  question 
raised  seems  to  be  :  *  Given  the  topic  induction,  what 
name  shall  be  given  to  the  science  that  includes  it  in 
its  treatment  ? '  rather  than  the  converse  question 
•  Given  the  name  logic,  shall  it  be  defined  so  as  to  in- 
clude, or  so  as  to  exclude,  induction  ?'  If  we  put  the 
question  in  the  first  form,  the  answer  is  of  course  purely 
arbitrary  ;  we  might  give  it  the  name  Epagogics.  But 
if  the  question  is  put  in  the  second  form,  the  answer  is 
not  in  the  same  sense  arbitrary,  assuming  that  there  is 
general  unanimity  as  regards  the  usage  of  the  name 
logic  to  denote  a  science  whose  central  or  essential 
function  is  to  criticise  thought  as  valid  or  invalid.  That 
induction  should  be  included  in  logic  thus  defined 
follows  from  the  undeniable  fact  that  we  do  infer  in- 
ductively, and  that  some  persons  in  reference  to  some 
problems  do  infer  invalidly.  Even  if  this  were  not  the 
fact,  it  is  certainly  of  scientific  importance  to  render 
explicit  what  everyone  implicitly  recognises  in  their 
inferences — as  much  for  the  case  of  induction  as  for 
that  of  syllogism  or  other  formal  types  of  inference. 
It  has  even  been  hinted  that  nobody  makes  mistakes 
in  formal  inference ;  and  yet — in  despite  of  this,  if 
true — no  one  questions  the  value  of  systematising  the 
principles  under  which  people  may  unconsciously  reason ; 
and  what  holds  of  formal  inference  would  certainly 
hold  a  fortiori  of  the  processes  of  inductive  inference 
which  present  many  more  serious  opportunities  for 
fallacy. 

§  2.    As  regards  the  term  *  thought '  which  enters 


into  my  definition,  its  application  is  intended  to  include 
R^iceptyal  judgments  which  are  commonly  contrasted 
with  rather  than  subsumed  under  thought,  for  the 
reason  that  thought  is  conceived  as  purely  abstract 
while  perception  contains  an  element  of  concreteness. 
But  properly  speaking  even  in  perceptual  judgment 
there  is  an  element  of  abstraction  ;  and  on  the  other 
hand  no  thought  involves  mere  abstraction.  It  follows, 
therefore,  that  the  processes  of  thinking  and  of  percep- 
tual judgment  have  an  essential  identity  of  character 
which  justifies  their  treatment  in  a  single  systematic 

,  whole.  It  is  the  distinction  between  sense-experience 
and  perceptual  judgment,  and  not  that  between  per- 

i  ceptual  judgment  and  thought,  that  must  be  emphasised. 
The  essential  feature  of  perceptual  judgment  in  con- 
trast to  mere  sense-experience  is  that  it  involves 
activity,  and  that  this  activity  is  controlled  by  the 
purpose  of  attaining  truth  ;  further  it  is  the  presence 
of  this  purpose  which  distinguishes  thought  from  other 
forms  of  mental  activity.  Thought  may  therefore  be 
defined  as  mental  activity  controlled  by  a  single  purpose, 
the  attainment  of  truth. 

§  3.  Now  it  is  true,  as  often  urged,  that  thought  is 
motived  not  solely  by  the  purpose  of  attaining  truth, 
but  rather  by  the  intention  of  realising  a  particular  end 
in  some  specific  form  and  under  certain  specific  cir- 
cumstances. But  I  have  to  maintain  that  any  other  or 
further  purpose  which  may  prompt  us  to  undertake  the 
activity  of  thinking  is  irrelevant  to  the  nature  of  thought 
as  such,  this  other  purpose  serving  only  to  determine 
the  direction  of  activity.  When  such  activity  is  actually 
in  operation  its  course  is  wholly  independent  of  the 


xvm 


INTRODUCTION 


INTRODUCTION 


XIX 


prompting  motive  and  guided  by  the  single  purpose  of 
attaining  truth.  For  instance,  our  desire  for  food  may 
prompt  us  to  search  for  it ;  but  this  resolve,  once  taken, 
leads  to  a  thinking  process  the  purpose  of  which  is  to 
come  to  some  conclusion  as  to  where  food  is  likely  to 
be  found,  and  the  sole  aim  of  this  process  is  to  discover 
what  is  true  on  the  matter  in  hand.  This  being  so,  the 
logical  treatment  of  thought  must  be  disencumbered 
from  all  reference  to  any  ulterior  purpose. 

Whether  truth  is  ever  pursued  without  any  ulterior  » 
purpose  is  a  psychological  question   which  may  fairly  \ 
be  asked ;  and  if  introspection  is  to  be  trusted  must » 
certainly  be  answered  in  the  affirmative,  although  the 
enquiry  whether  true  knowledge  has  intrinsic  value  or 
not  belongs  to  ethics.  That  the  attainment  of  truth  for 
its  own  sake  constitutes   a  genuine  motive   force  is 
further  confirmed  by  recognising  the  fact  that  people 
do   actually    attach    value    to    true    knowledge,   as  is 
incontestably  proved  by  their  willingness  to  defy  the 
prospect  of  social  disapprobation,  persecution,  and  even 
martyrdom  incurred  by  the  utterance  and  promulgation 
of  what  they  hold  to  be  true.  At  the  same  time,  it  must 
be  pointed  out  that  the  aim  of  the  thinking  process  is 
not  the  attainment  of  truth  in  general,  but  always  of 
truth  in  regard  to  some  determinate  question  under 
consideration.  This  is  closely  analogous  to  the  psycho- 
logical fact  that  what  we  desire  is  never  pleasure  in 
general,  but  always — if  the  doctrine  of  psychological 
hedonism  is  to  be  accepted — some  specific  experience 
which  is  represented  as  pleasurable. 

Any  thinking  process  is  normally   initiated  by  a 
question   and   terminated   by   an   answer;   what   dis- 


tinguishes one  thinking  process  from  another  is  the 
difference  of  the  question  proposed.  The  bond  of  unity 
amongst  the  phases  of  a  single  process  does  not 
necessarily  entail  unbroken  temporal  continuity,  but 
only  identity  of  the  question  proposed.  Indeed  any 
thought  process  may  be  temporarily  interrupted  before 
the  proposed  question  has  been  answered.  It  must  be 
left  as  a  topic  for  psychology  to  investigate  the  causes  of 
such  suspension,  and  how  far  the  advance  made  serves 
as  a  starting  point  for  further  advances.  Logic,  on  the 
other  hand,  is  concerned  with  the  nature  of  the  advance 
as  an  advance  and  criticises  the  process  from  the  point 
of  view  of  validity  or  invalidity. 

§  4.  The  above  definition  of  logic  as  the  analysis  and 
criticism  of  thought  should  be  compared  with  that  of 
the  Scholastics,  who  laid  emphasis  on  the  point  that 
logic  is  concerned  with  the  art  of  thinking,  where  art 
is  nearly  equivalent  to  the  modern  term  technique,  and 
has  an  understood  reference  to  activity  with  an  end  in 
view.  The  study  of  the  art  of  thinking  as  thus  under- 
stood is  of  use  in  instructing  us  how  to  proceed  when 
thinking  out  any  problem :  for  instance,  it  lays  down 
rules  of  classification  and  division  for  the  clearing  up 
of  obscurities  and  inconsistencies  in  thought ;  rules  for 
the  recall  and  selection  of  knowledge  appropriate  to 
any  given  problem ;  etc.  Descartes'  Discourse  on 
Method  is  a  classical  illustration  of  this  species  of 
science.  Modern  examples  of  excellent  treatises  on  these 
lines  are  to  be  found  in  Alfred  Sidgwick,  and  other 
neo-pragmatists.  It  is  a  science  of  the  highest  value, 
and  need  only  be  separated  from  logic  on  the  ground 
of  the  difference  of  purpose;  inasmuch  as  its  direct 


XX 


INTRODUCTION 


INTRODUCTION 


XXI 


purpose  IS  the  aUaini^^  of  valid  thought,  whereas 
logic  is  the  study  of  the  conditions  of  valid  thought, 
and  as  such  it  does  not  exclude  the  study  of  the  art. 

§  5.  Alongside  of  the  use  of  the  term  *art'  to  mean 
technique,  there  is  a  more  modern  usage  where  it 
implies  reference  to  aesthetic  feelings  and  judgments. 
Nowadays  discussions  as  to  whether  an  objective 
standard  for  these  feelings  and  judgments  should  be 
recognised  are  very  prominent.  The  nature  of  the 
feelings  and  judgments  that  enter  into  aesthetic 
appreciation  belongs  to  psychology  ;  but  if  we  agree 
that  there  is  a  discoverable  objective  standard,  then 
the  treatment  of  the  subject  of  aesthetics  is  to  be 
distinguished  from  the  psychological  treatment,  precisely 
as  the  treatment  of  thought  in  logic  is  distinguished 
from  that  in  psychology. 

Aesthetics,  in  this  sense,  raises  very  similar  problems 
to  those  presented  in  Ethics ;  and  it  is  frequently  said 
that  as  normative  Logic,  Aesthetics  and  Ethics  are 
related  in  the  same  way  to  the  three  psychological 
factors,  thought,  feeling  and  volition  respectively.  Each 
of  the  normative  studies  may  be  said  to  be  based  on  a 
standard  of  value,  the  precise  determination  of  which 
it  is  their  function  to  formulate;  in  each,  imperatives 
are  laid  down  which  are  acknowledged  by  the  in- 
dividual, not  on  any  external  authority,  but  as  self- 
imposed ;  and,  in  each,  the  ultimate  appeal  is  to  the 
individual's  intuitive  judgment.  There  is,  however,  a 
closer  resemblance  between  Ethics  and  Aesthetics  in 
their  relations  to  volitions  and  feelings  respectively, 
than  between  either  of  them  and  Logic ;  inasmuch  as 
there  are  apparently  fundamental  differences  of  opinion 


as  to  the  ultimate  ethical  and  aesthetical  standards,  that 
give  to  the  studies  of  Ethics  and  Aesthetics  a  con- 
troversial character  absent  from  Logic  about  whose 
standards  there  is  no  genuine  disagreement.  As  regards 
the  relation  of  Ethics  to  Logic,  the  question  sometimes 
arises  as  to  which  subject  is  supreme.  The  answer 
to  this  question  depends  entirely  upon  the  nature  of 
the  supremacy  intended:  the  imperatives  for  thought 
become  imperatives  for  conduct  only  on  condition  that 
true  judgments  have  intrinsic  value  and  false  judgments 
intrinsic  disvalue ;  and  thus,  from  the  point  of  view  of 
conduct,  Logic  is  subordinate  to  Ethics.  On  the  other 
hand,  ethical  enquiry — like  any  other  scientific  investi- 
gation— has  to  avoid  violating  logical  principles,  so 
that  from  the  point  of  view  of  true  thought  Logic  is 

I  supreme  over  Ethics. 

§  6.  Our  discussion  so  far  has  led  us  to  consider  the 
relations  of  Logic  to  Philosophy  in  general.  Psychology, 
Aesthetics  and  Ethics.  Another  subject  to  which  it  is 
closely  allied  and  from  which  it  is  yet  distinct  is 
Grammar,  the  alliance  h^xng  prima  facie  accounted  for 
by  the  common  concern  of  the  two  studies  with 
language.  The  connection  between  thought  and  language 
presents  a  problem  for  the  science  of  Psychology ;  but, 
so  far  as  thinking  or  the  communication  of  thought  in- 
volves the  use  of  words,  the  provinces  of  Logic  and 

s  Grammar  coincide ;  that  is  to  say  universal  Grammar, 
which  excludes  what  pertains  to  different  languages 
and  includes  only  what  is  common  to  all  languages, 
should  be  subsumed  under  Logic.  For  the  modes  in 
which  words  are  combined — which  constitute  the  sub- 
ject matter   for  Grammar — cannot  be  expounded  or 


/•K 


a..  I 


xxu 


INTRODUCTION 


INTRODUCTION 


XXIU 


understood  except  as  reflecting  the  modes  in  which 
thoughts  are  combined ;  and  this  combination  is  effected 
by  means  of  such  logical  operations  as  negation,  con- 
junction, disjunction,  alternation,  implication  and  so  on, 
represented  by  the  words  not,  and,  not  both,  or,  if,  etc. 
To  justify  the  subordination  of  Grammar  to  Logic  we 
have  only  to  realise  that  the  analysis  of  the  sentence 
in  Grammar  corresponds  to  the  analysis  of  thought  in 
Logic,  and  that  grammatical  criticism  is  confined  to 
securing  that  the  sentence  precisely  represents  the 
thought,  any  further  criticism  of  the  proposition  coming 
exclusively  within  the  province  of  Logic.  It  may  be 
pointed  out  in  this  connection  as  specially  significant 
both  for  the  linguist  and  for  the  logician,  that  languages 
differ  in  the  degree  of  their  capacity  to  exhibit  through 
their  structure  intimacy  between  words  and  thoughts. 

§  7.  Amongst  all  the  sciences  over  which  logic  must 
rule,  there  is  one  that  occupies  a  unique  place.  The 
constituents  of  thought  which  are  in  the  most  narrow 
sense  logical  are  those  which  give  form  to  the  construct, 
connecting  alien  elements  by  modes  which  give  specific 
significance  to  the  whole.  The  first  group  of  these  is 
expressed  by  ties,  conjunctional  words,  prepositional 
words,  and  modes  of  verbal  inflection.  But  as  the  form 
of  thought  is  further  elaborated  there  enter  new  kinds 
of  terms,  namely  specific  adjectives  which  have  a  con- 
stant meaning  definable  in  terms  of  pure  thought,  or 
else  are  to  be  admitted  and  understood  as  indefinables. 
The  most  generic  form  of  such  adjectives  directly 
expresses  the  result  of  such  mental  acts  of  comparison 
as  like,  unlike,  different  from,  agreeing  with.  Owing  to  i 
the  purely  logical  nature  of  these  relations,  universal 


formulae  in  which  theyare  introduced  can  be  constructed 
by  mere  abstract  thought.  The  preliminary  condition 
for  this  construction  is  the  separating  of  what  is  given 
to  constitute  a  plurality,  and  thus  to  introduce  a  formal 
factor  which  can  only  be  verbally  expressed  by  the 
separations  and  juxtapositions  of  the  substantial  words. 
i  The  very  general  relation  that  separation  effects  is 
1  that  most  indeterminate  relation  otherness.  When  the 
complementary  notions  of  separateness  and  together- 
ness are  joined  to  constitute  a  unity,  there  enters  the 
idea  of  number,  and  we  are  in  the  domain  of  mathe- 
matics. 

The  extraordinary  capacity  for  development  that 
marks  mathematics  is  due  to  the  precision  with  which 
the  relations  of  comparison  are  capable  of  being 
amplified.  Through  the  substitutions  that  are  thus 
rendered  possible,  the  range  of  application  of  mathe- 
matical formulae  is  extended  beyond  the  bounds  which 
would  otherwise  delimit  logic.  Any  material  that  might 
be  presented  to  thought  upon  which  the  same  precise 
operations  of  comparison  could  be  performed,  would 
lead  to  the  same  forms  as  mathematics.  For  example 
ideas,  not  only  of  difference,  but  of  determinable 
degrees  of  difference,  bring  the  material  into  relations 
of  intrinsic  order,  and  out  of  these  relations  emanate 
relations  between  relations,  so  that  theoretically  the 
science  develops  into  a  highly  complicated  system.  The 
point  then,  where  we  may  venture  to  say  that  logic 
actually  passes  into  mathematics  is  where  the  specific 
indefinable  adjectives  above  referred  to  give  new 
material  for  further  logical  combinations. 

Here  it  is  of  great   importance  to  point  to  the 


f Ov^  31 


xxiv  INTRODUCTION 

relative  nature  of  the  distinction  between  form  and 
matter.  Logic  begins  with  a  sharp  contrast  between 
matter,  as  what  is  given  as  merely  shapeless,  and 
form,  as  that  which  thought  imposes.  But  as  we 
advance  to  mathematics,  we  impose  a  new  element  of 
form  in  introducing  the  relation  otherness  and  its 
developments ;  and  this  being  operated  on  by  thought 
takes  the  place  of  new  matter:  in  short,  what  is 
introduced  as  matter  is  form  in  the  making.  All  this 
could  be  summed  up  by  saying  that  for  elemental  logic, 
mathematical  notions  would  constitute  matter ;  whereas 
when  the  step  into  mathematics  is  once  taken  these 
same  elements  are  just  those  in  accordance  with  which 
thought  advances  in  constructing  more  and  more  com- 
plicated forms.  This  view  of  the  relation  of  logic  to 
mathematics  will  be  worked  out  in  Part  II  of  the 
present  work  under  'Demonstration,'  where  the  pro- 
cedure of  building  up  mathematical  science  is  shown  to 
involve  the  very  same  principles  as  are  used  in  the 

logical  structure. 

All  the  sciences,  including  mathematics,  over  which 
logic  has  supreme  control,  have  been  properly  described 
as  applied  logic.  But  mathematics  is  applied  logic  in  a 
certain  very  unique  sense,  for  mathematics  is  nothing 
but  an  extension  of  logical  formulae  introducing  none 
but  purely  logical  factors ;  while  every  other  science 
borrows  its  material  from  experiential  sources,  and 
can  only  use  logical  principles  when  or  after  such 
material  is  supplied.  Within  mathematics  we  have 
again  the  same  kind  of  distinction,  namely  that  between 
pure  and  applied  mathematics,  as  it  has  been  called. 
In  pure  mathematics,  the  mathematician  can  give  free 


INTRODUCTION 


XXV 


play  to  his  imagination  in  constructing  forms  that  are 
restricted  only  by  principles  of  logical  consistency,  and 
he  develops  the  implications  that  are  derivable  from 
what  may  be  indifferently  regarded  either  as  definitions 
of  his  fictitious  constructs  or  as  hypothetically  enter- 
tained first  axioms.   In  order  that  these  axioms  and  the 
theorems  therefrom  derived  may  be  considered  as  true, 
recourse  must  be  had  to  the  real  world,  and  if  applicable, 
the  axioms  come  to  be  assertorically  entertained  as 
premisses,  and  the  derived  propositions  as  the  developed 
conclusions.   This  application  of  mathematics  to  reality 
constitutes  applied  mathematics.    Taking  geometry  as 
our  first  example,  while  there  is  no  limit  to  constructing 
conceived  spaces  other  than  Euclidian,  their  application 
to  reality  demands  the  enquiry  whether  our  space  is  or 
is  not  Euclidian.   This  is  answered  by  an  appeal  to  our 
immediate  intuitions  directed  to  our  spatial  experiences, 
and  it  is  this  appeal  that  is  outside  the  range  of  pure 
mathematics.    Again  the  merely  logical  conception  of 
betweenness,  which  develops  into  that  of  serial  orders 
of  lower  or  higher  forms  of  complexity,  is  in  the  first 
instance  a  product  of  pure  logical  constructiveness,  and 
would    yield    implications    from    which    a    system    of 
implicates  could  be  developed.     But  such  a  hypotheti- 
cally conceived  body  of  propositions   would  have  no 
basis  in  the  real  but  for  the  applicability  of  the  defined 
conceptions    to    what    is   given    in    non-mathematical 
intuition.     This    applicability    holds    not   only    in   the 
domain    of    spatial    order,    but   also    in    that    of    the 
qualitative  relations  of  difference  which  impose  serial 
order  amongst  sense  impressions. 

Regarded  in  the  light  of  its  control  over  all  sciences 

J.L.  ^ 


xxvi 


INTRODUCTION 

logic  has  been  called  by  the  name  'Methodology'; 
that  is  to  say  while  the  forms  of  logic  implicitly  control 
the  conclusions  of  science,  logic  itself  includes  the  study 
which  renders  explicit  the  ways  according  to  which  its 
authority  is  exercised.  The  department  of  logic  known 
as  methodology  constitutes  the  third  part  of  the  present 
work,   which   is  entitled   'The  logical   foundations  of 

Science.' 

Another  illustration  of  applied  mathematics  is  to 
quantity.    Quaruity  is  not  a  mere  direct  development 
from  number,  since  a  new  conception,  namely  that  ot 
equality  of  units,  enters  as  a  distinctive  factor  which  is 
not  purely  logical.     It  is  true  that  equality  for  merely 
formal  developments   could  be   defined   as  a  certain 
relation  having  the  formal  properties  of  symmetry  and 
transitiveness,  and  if  to  this  conception  is  added  the 
fundamental  operation  plus  (  + ),  definable  as  a  certain 
relation  having  the  formal  properties  commutative  and 
associative,  the  whole  system  of  quantitative  science 
could  be  developed  without  recourse  to  any  but  pure 
mathematical  principles.    But   even  in  this  range  of 
thought    quantities    of    different   types    would    need 
recognition.    For  example,  given  the  notion  of  length 
as  the  first  spatial  quantity,  a  new  quantity  is  derived 
by  multiplying  length  by  length,  which  is  called  area; 
here  'multiplied'  need  not  be  more  specifically  defined 
than  a  certain  relation  having  the  formal  properties 
commutative  and  associative.    Again  where  a  quantum 
of  space  is  divided  by  a  quantum  of  time,  we  have 
velocity,  and  in  this  way  a  totally  new  type  of  quantity 
is  constructed  and  we  pass  from  geometry  to  kinematics. 
Another  quantity  called  mass  is  such  that  when  multi- 


INTRODUCTION 


xxvu 


plied  by  velocity  there  is  engendered  the  new  quantity 
called  momentum,   and   when  multiplied    by   velocity 
squared,  energy ;  and  in  the  introduction  of  these  new 
species  of  quantity  we  pass  from  kinematics  to  dynamics. 
This  is  the  terminus  on  these  lines  of  applied  mathe- 
matics; and  dynamics  may  be  defined  as  the  science 
that  uses  the  three  independently  definable  species  of 
quantity  time,  space  and  mass.     In  every  extension, 
then,  of  mathematics  no  new  idea  or  mode  of  thought 
need  accompany  the  work  of  the  calculus.    It  is  only 
when  the  formulae  have  to  be  applied  to  reality,  and 
thus  to  be  entertained  categorically,  that  a  process  of 
thought  other  than  merely  mathematical  enters  in,  and 
intuition  is  directed  to  what  is  given  in  some  form  of 
experience.  The  ideas  which  enter  in  to  the  mathematical 
sciences  thus  constructed  have  a  form  which  renders 
them  amenable  to  purely  logical  processes  of  indefinite 
degrees  of  complexity;  this  distinguishes  them  from 
the  non-mathematical  or  'natural'  sciences  that  intro- 
duce   ideas    dependent    simply    upon    brute    matter, 
unamenable  to  logical  analysis,  logic  entering  only  in 
the  application  to  these  ideas  of  classification,  and  the 
principles  of  inductive  inference. 

§  8.  Having  considered  logic  in  its  relation  to  the 
different  sciences,  we  may  now  pass  to  a  discussion  of 
its  more  philosophical  aspects.  Logicians  have  been 
classified  as  nominalists,  conceptualists,  and  realists  or 
materialists,  according  as  they  think  it  worth  while  to 
I  discuss  words,  thoughts  or  things.  Names  that  are  apt 
to  be  understood  as  synonyms  for  these  have  been 
I  applied  to  different  philosophical  opinions;  and  this 
fact  is  indicative  of  the  change  which  has  occurred  in 


C2 


xxviii 


INTRODUCTION 


INTRODUCTION 


XXIX 


the  course  of  the  history   of  philosophy,   where  the 
ground  has  been  shifted  from  ontology  to  psychology, 
and  later  from  psychology  to  logic.    To  take  realism 
first.     It  is  the  name  given  to  the  Platonic  view  which 
formed  the  basis  of  Aristotle's  controversy  with  Plato. 
Plato  in  discussing  the  relation  between  the  universal 
and  the  individual,  attributes  rm/ existence  in  the  truest 
or  most  ultimate  sense  to  the  universal,  holding  that 
the  particular  individual  has  reality  only  so  far  as  it 
partakes  of  the  nature  of  the  universal,  towards  which 
it  strives  as  the  end  (ivTeXeKT))  of  its  existence.  Aristotle, 
opposing  this  view,  holds  that  the  universal  exists  not 
apart  from  the  particular  but  in  it. 

A  new  psychological  significance  came  to  be  attached 
to  the  term  Realism,  when  the  question  of  reality  was 
raised  not  about  the  MzW^,  but  about  the  possible  idea 
of  the  thing,  these  two  concepts  being  taken  to  be 
equivalent.    The  so-called  nominalist  school  of  philo- 
sophers maintained  the  psychological  view  that  we  had 
no  idea  corresponding  to  a  general  name,  along  with 
the  ontological  view  according  to  which  the  particular 
individual  or  concrete  alone  existed,  and  no  existence 
could  be  attributed  to  the  universal ;   generality,   for 
them,    attached   only   to  names  in  use,   and  had    no 
objective  application.    On  the  psychological  point  at 
issue  the  opponents  of  this  view  have  been  known  as 
I  conceptualists,  and  in  maintaining  their  opposition  were 
led  to  make  a  psychological  distinction  of  great  im- 
portance between  images  and  ideas.    In  common  with 
the  nominalists,  they  held  that  images  are  necessarily 
concrete,  particular  or  individual,  but  they  maintained 
that  we  can  also  frame  ideas  which  can  properly  be 


called  abstract  or  general.  Both  schools  assumed  that 
images  were  equivalent  to  or  at  least  resembled  per- 
ceptions, and  further  that  the  latter  were  obviously 
concrete  and  particular.  Berkeley  represents  the  nomi- 
nalist school,  and  his  subtle  difference  from  Locke — 
who  definitely  held  that  we  can  frame  general  ideas, 
though  with  difficulty — comes  out  clearly  when  he  dis- 
putes the  possibility  of  a  general  idea  of  a  triangle 
(instanced  by  Locke)  which  shall  be  neither  equilateral 
nor  isosceles  nor  scalene,  and  from  which  we  can  in 
thought  abstract  the  shape  from  variations  of  colour. 
In  my  view  Locke  and  Berkeley  were  both  wrong,  ^ 
even  where  they  agreed  ;  inasmuch  as  neither  images 
nor  perceptions  reflect  the  concreteness  and  particularity 
of  the  individual  thing,  which  should  be  described  as 
determinate,  in  contrast  to  the  indeterminateness  of  the 
mental  processes.  In  fact  there  has  been  a  confusion 
in  the  description  of  our  thoughts,  images  and  percepts, 
between  the  distinction  of  the  universal  from  the  3 
particular,  and  that  of  the  indeterminate  from  the  de-  ^. 
terminate.  The  modern  term  'generic,'  which  has 
been  applied  to  images,  should  be  extended  also  to 
percepts,  on  the  ground  that  they  share  with  images 
the  character  of  indeterminateness — a  character  which 
must  be  rigidly  distinguished  from  general  or  uni- 
versal as  properly  applied  to  ideas  or  concepts. 

Nominalism  has  yet  another  meaning  when  applied 
as  a  special  logical  theory  ;  in  this  sense  it  denotes  the 
theory  according  to  which  the  proposition  is  an  indica- 
tion of  the  names  that  have  been  arbitrarily  chosen  to 
denote  things  or  classes  of  things,  and  predicates  merely 
what  follows  from  the  consistent  use  of  these  names. 


^ 


XXX 


INTRODUCTION 


INTRODUCTION 


XXXI 


Propositions  are  thus  used  as  mere  formulae  and  re- 
peated in  thought  when  necessary,  without  demanding 
any  consideration  of  their  meaning ;  so  that  the  only 
ultimate   foundations  or  premisses  of  knowledge  are 
definitions,  no  other  propositions  of  the  nature  of  axioms 
being  required.    This  view  still  clings  to  some  modern 
philosophical  expositions  of  arithmetic  and  pure  logic,  and 
is  rather  subtly  akin  to  the  view  that  the  first  premisses 
for  science  are  nothing  but  postulates  or  hypotheses 
which,  if  consistently  held,  lead  to  the  discovery  of  truth. 
As  regards  Conceptualism,  it  is  doubtful  whether, 
as  applied  to  the  work  of  such  writers  as  Hamilton  and 
Sigwart,  it  can  be  properly  regarded  as  a  distinctive 
logical  theory.    For  the  prominent  use  of  the  word 
concept  and  its  associate  judgment  points  not  neces- 
sarily to  any  difference  of  logical  theory  between  those 
who  use  these  words,  and  those  who  prefer  the  words 
*term'  or  *name'  and  ^proposition,'  but  merely  to  the 
common    recognition  that  thought  has  form  as  well 
as  verbal  expression.    If,  however,   the  conceptualist 
proceeds  to  limit  the  scope  of  logic  to  the  consideration 
of  the  forms  of  thought  alone,  then  he  must  maintain 
that  the  truth  of  a  judgment  is  tested  by  the  form  that 
connects  the  content  as  conceived  ;  and  conceptualism 
becomes  equivalent  to  formalism.  The  criterion  for  the 
=  formalist  is  indeed  mere  consistency  or  coherence  in 
fact ;  that  for  the    conceptualist  proper,   clearness  or 
I  distinctness    in    thought.      The    latter    is    expressed 
negatively  by  Herbert  Spencer  :  what   is  clearly  not 
conceivable  is  false ;  positively  by  Descartes  :  what  is 
clearly  conceivable  is  true.   It  follows  immediately  from 
this  view  that  truth  concerns  only  conceived  content ; 


so  that  the  direct  objects  of  thought  are  not  things, 
but  our  ideas  about  things,  and  judgment  contains  no 
\  reference  to  things  but  only  to  adjectives.  On  this 
understanding,  the  conceptualist's  view  is  that  we  can 
only  deal  with  things  as  conceived,  and  that  it  is  the 
mode  under  which  we  conceive  them  that  determines 
the  adjectives  themselves  and  their  relations  as  consti- 
tuting the  content  of  the  judgment.  In  this  way  they 
are  led  to  deny  all  relations  as  subsisting  between 
things — a  denial  which  is  simply  equivalent  to  denying 
the  one  supreme  relation  otherness ;  for  otherness  may 
be  said  to  be  the  one  determinable  relation  to  which  all 
specific  relations  stand  as  determinates.  Hence  it  is 
enough  for  this  school  of  philosophy  to  deny  the  single 
relation  otherness,  and  in  this  denial  to  adopt  the 
position  of  monism.  The  view,  if  carried  out  rigidly, 
goes  beyond  that  of  Spinoza,  who  asserted  that  thought 
was  other  than  extension,  and  even  that  the  one 
Substance  had  an  infinity  of  other  attributes,  though 
not  conceivable  by  us.  It  is  an  odd  fact  that  Lotze,  in 
particular,  explicitly  rejects  relations  only,  as  expressive 
of  the  nature  of  Reality  ;  but  in  consistency  he  ought ' 
to  have  included  in  his  rejection  ordinary  adjectives.! 
From  this  point  of  view,  the  only  kind  of  singular 
categorical  judgment  concerns  Reality  as  a  whole  and 
not  any  one  of  its  several  separable  parts  :  it  predicates 
character  of  the  indivisible  one,  not  of  this  or  that  unit 
in  the  one.  Individual  units,  in  fact,  are  conceived  as 
the  result  of  the  imposition  of  thought  to  which  nothing 
in  the  one  corresponds.  Thus  the  Monist's  first  principle 
is  to  deny  the  Pluralisms  fundamental  assumption  that 
the  Real,  as  given  to  thought,  is  given  as  many  and  as 
such  involves  existential  otherness. 


/ 


U^ 


xxxii 


INTRODUCTION 


INTRODUCTION 


XXXUl 


The  conceptualist's  account  of  the  character  of  the 
singular  judgment  leads  to  a  similar  account  of  that 
of  the  particular  and  the  universal  judgment.  The 
view  is  consistently  borne  out  by  his  interpretation  of 
particulars  as  possible  conjunctions,  i.e.  of  adjectives 
that  we  can  conjoin  in  conception ;  and  of  universals  as 
necessary  conjunctions,  i.e.  of  adjectives  that  we  must 
conjoin  in  conception.  Symbolically:  '  Some  things  that 
are  /are  ^'  is  to  mean  'p  and  q  can  be  conjoined  and  can 
be  disjoined' ;  '  Everything  or  nothing  that  is/  is  q '  is  to 
mean  'p  and  q  must  be  conjoined  or  must  be  disjoined.' 
What  is  true  in  this  view  is  that  the  operations  not,  and, 
not-both,  if,  or,  are  supplied  by  thought ;  and  that  nothing 
in  the  merely  objective  world  manifests  the  mere  absence 
of  a  character,  or  the  mere  indeterminateness  of  the 
alternative  operation,  or  dependence  as  expressed  by  im- 
plication. These  relations  are  not  manifested  to  thought, 
but  analytically  or  synthetically  discovered  or  rather  im- 
posed by  thought.  The  view  is  most  strikingly  expressed 
by  Mr  Bradley  in  his  dictum:  only  if  what  is  possible  is 
necessitated  will  it  be  actualised;  and  again,  only  if 
what  is  necessary  is  possible  will  it  be  actualised. 

From  conceptualism  we  pass  back  again  to  realism 
in  its  new  sense  as  applying  to  logic,  and  in  this  appli- 
cation it  is  usually  denoted  by  the  term  materialism  or 
empiricism.    We  are  thus  led  back  again  to  Venn,  and 
less  explicitly  to  Mill,  who  contrasts  the  formalism  or 
conceptualism  of  Hamilton  with  his  own  logical  stand- 
point.   Taking  empiricism  to  mean  that  all  knowledge  j 
is  obtained  by  experience  alone  (as  Mill  only  seems  to  | 
have  held)  the  doctrine  amounts  to  maintaining  that  all  \ 
inference  is  ultimately  of  the  nature  of  pure  induction.  I 
But  taking  it  to  mean  that  no  knowledge  gained  by 


experience  can  be  validly  universalised  (as  Venn  seems 
to  hold)  then  the  doctrine  amounts  to  maintaining  that 
no  inference  of  the  nature  of  pure  induction  is  valid,  and 
that  hence  only  deduction  is  guaranteed  by  logic.    In 
default  of  any  explication  of  which  of  these  two  views 
is  meant  by  empiricism  or  materialism,  we  can  only 
conclude  that  the  term  stands  for  that  department  of 
logic  that  is  concerned  with  an  analysis  of  the  process 
of  induction.    But  here  we  must  note  that  the  distinction 
in  character  between  induction  and  deduction   is  not 
properly  expressed  by  the  antithesis  of  matter  and  form; 
since  the  relations  amongst  premisses  and  conclusion 
which  constitute  the  form  of  an  inference  hold  for  the 
validity  of  induction  as  for  that  of  deduction ;  and  con- 
versely, reference  to  the  matter  of  the  propositions  is 
required  equally  for  the  truth  of  a  deductive  inference 
as  for  that  of  an  inductive  inference.    This  obvious  fact 
has  been  forgotten,   owing  to  the  great  prominence 
given  by  inductive  logicians  to  the  treatment  of  the 
preliminary  processes  of  observation,  search,  arrange- 
ment, comparison  of  material  data,  and  the  formation 
of  formulae  that  shall  hold  for  the  facts  collected,  and 
the  aid  required  by  experimentation.    In  consequence, 
stress   is   laid  on   the  securing  of  correctly  described 
premisses  in  the  case  of  induction ;  whereas  in  the  case 
of  deduction  stress  is  laid  only  on  securing  validity  for 
the  form  of  inference. 

§  9.  In  conclusion  I  propose  to  enumerate  the  most 
important  features  in  the  treatment  of  logical  theory  to 
be  developed  in  the  course  of  this  work: 

{a)  The  epistemic  aspect  of  thought  is  included 
within  the  province  of  logic,  and  contrasted  with  the 
constitutive  aspect ;    the  former  is  a  recognition   that 


If. 


XXXIV 


INTRODUCTION 


INTRODUCTION 


XXXV 


-^/^ 


knowledge  depends  upon  the  variable  conditions  and 
capacities  for  its  acquisition  ;    the  latter  refers  to  the 
content  of  knowledge  which  has  in  itself  a  logically 
analysable  form.   Such  fallacies  2.^ petitio principii  really 
require  reference  to  the  epistemic  aspect  of  thought, 
while  fallacies  of  the  strictly  formal  type  refer  exclu- 
sively to  the  constitutive  aspect.  Again  the  whole  theory 
of  modality  which  develops  into  probability  is  essentially 
epistemic,  indicating  as  it  does  the  relation  of  the  con- 
tent of  the  proposition  to  the  thinker.    Thus  a  distinc- 
tion is  clearly  drawn  between  the  proposition  and  the 
attitude  of  assertion  or  judgment ;  and  while,  on  this 
view,  the  proposition  is  identifiable  when  in  variable 
relations  to  different  thinkers,  the  necessity  is  empha- 
sised of  conceiving  the  proposition  in  terms  of  assertion, 
the  act  of  assertion  being  thus  taken  as  the  complete 
fact  to  be  analysed  and  criticised.    It  is  this  intimate 
connection  between  the  assertion  and  the  proposition 
which  gives  meaning  to  the  identification  of  the  adjec- 
tives true  and  false  with  the  imperatives  'to  be  accepted' 
and  'to  be  rejected.' 

{b)  The  proposition  itself,  which  is  customarily  re- 
solved into  subject  and  predicate,  is  more  precisely 
analysed  by  showing  that  the  substantive  alone  can 
function  as  subject,  and  the  adjective  as  predicate,  and 
that  these  stand  to  one  another  in  the  relation  of 
characterisation :  the  substantive  being  that  which  is 
characterised,  the  predicate  that  which  characterises. 
Since  an  appropriate  adjective  can  be  predicated  of  a 
subject  belonging  to  any  category,  including  adjective, 
relation  and  proposition,  the  subject  as  thus  functioning 
becomes  a  quasi-substantive.  The  substantive  proper 
seems  to  coincide  with  the  category  *  existent,'  while 


jL 


\f^\ 


if  any  category  other  than  substantive  stands  as  subject 
its  logical  nature  is  not  thereby  altered,  but  rather  the 
adjectives  proper  to  it  fall  under  correspondingly  special 
sub-categories  determined  by  the  category  to  which  the 
subject  belongs. 

{c)  Adjectives  are  fundamentally  distinguishable 
into  determinates  and  determinates,  the  relation  be- 
tween which  is  primarily  a  matter  of  degree,  a  deter- 
minable being  the  extreme  of  indeterminateness  under 
which  adjectives  of  different  degrees  of  determinateness 
are  subsumed.  The  relation  of  a  determinate  to  its 
determinable  resembles  that  of  an  individual  to  a  class, 
but  differs  in  some  important  respects.  For  instance, 
taking  any  given  determinate,  there  is  only  one  deter- 
minable to  which  it  can  belong.  Moreover  any  one 
determinable  is  a  literal  summum  genus  not  subsumable 
under  any  higher  genus ;  and  the  absolute  determinate 
is  a  literal  infima  species  under  which  no  other  deter- 
minate is  subsumable. 

{d)  Relations  are  treated  as  a  specific  kind  of  ad- 
jective, and  are  called  transitive  adjectives  in  distinction 
from  ordinary  adjectives  which  are  intransitive.  The 
adjectival  nature  of  relations  is  apt  to  be  obscured  by 
the  inclusion  under  relative  terms  of  what  are  merely 
substantives  defined  by  relational  characterisation.  All 
that  holds  universally  of  adjectives,  including  the  rela- 
tion of  determinates  to  their  determinable,  holds  of 
relations  as  such. 

{e)    Under  the  head  of  induction,   fundamentally 
different  types  are  distinguished.    First :  the  very  ele-    -^* 
mentary  process  of  intuitive  induction,  which  lies  at  the   f 
basis  of  the  distinction  between  form  and  matter,  and 
by  which  all  the  formal  principles  of  logic  are  estab- 


XXXVl 


INTRODUCTION 


INTRODUCTION 


xxxvu 


lished.  Next:  summary  induction,  more  usually  called 
perfect  induction,  which  establishes  conclusions  of  limited 
universality  by  means  of  mere  enumeration.  Such  a 
summary  universal  stands  as  premiss  for  an  unlimited 
universal  conclusion,  obtained  by  what  is  called  indue- 
tio  per  simplicem  enumerationem.  What  is  specially 
important  in  my  treatment  is  the  function  of  summary 
induction  in  the  specifically  geometrical  form  of  infer- 
ence. Thirdly :  demonstrative  induction,  which  employs 
no  other  principles  than  those  which  have  been  recog- 
nised in  deduction.  This  species  of  induction  is  directly 
employed  in  inferring  from  a  single  experimental  in- 
stance an  unlimited  universal ;  and  it  is  this  species  of 
induction  which  gives  the  true  form  to  the  methods 
formulated  by  Mill  and  Bacon.  Lasdy  we  distinguish 
induction  proper,  which  is  conceived  as  essentially 
problematic,  and  as  thus  re-introducing  the  epistemic 
aspect  in  the  form  of  probability. 

(/) .  The  specific  notion  of  cause  as  applying  to 
events  is  distinguished  from  the  generic  notion  of  mere 
determination  according  to  a  universal  formula.  As 
specific,  cause  relates  exclusively  to  states  or  conditions 
temporally  alterable  and  also  referable  to  place ;  and,  in 
this  application  of  the  notion  of  determination,  the  effect 
and  cause  are  homogeneous.  Not  only  is  the  character 
of  the  effect  regulated  by  that  of  the  cause,  but  the  date 
and  place  of  the  latter  is  determined  by  the  date  and 
place  of  the  former.  The  universal  positional  relation, 
as  it  may  be  called,  of  cause  to  effect  is  that  of  con- 
tiguity, which  is  to  be  conceived  in  the  form  of  the 
coincidence  of  the  temporal  or  spatial  boundary  of  that 
which  constitutes  the  cause  with  that  which  constitutes 
the  effect.    This  absolute  contiguity  disallows  any  gap 


between  the  cause  process  and  the  effect  process;  so 
that  contiguity  is  strictly  defined  as  equivalent  to  con- 
tinuity. This  further  implies  that  when,  as  is  always 
permissible,  we  conceive  a  phase  of  the  causal  process 
as  temporally  or  spatially  separated  from  a  phase  of  the 
effect  process,  we  must  also  conceive  of  that  which  goes 
on  in  the  interval  bridging  cause  and  effect  to  be  part 
of  one  continuous  process.  This  is  possible  because 
time  and  space  are  themselves  continuous.  Thus 
change  and  movement  are  connectionally  continuous, 
in  the  special  sense  that  the  character  manifested  at  one 
instant  of  time  or  at  one  point  of  space  differs  from  that 
manifested  at  another  instant  of  time  or  at  another 
point  of  space,  in  a  degree  the  smallness  of  which 
depends  upon  that  of  the  temporal  or  spatial  interval. 
Again  superimposed  upon  the  continuity  of  this  process, 
there  is  a  discontinuity  of  the  second  order,  ultimately 
due  to  the  discontinuous  occupation  of  space  by  different 
kinds  of  matter. 

X,  (^)  The  notions  of  cause  and  substance  reciprocally 
imply  one  another,  the  latter  being  that  which  continues 
to  exist  and  in  which  alterable  states  or  conditions 
inhere.  These  alterable  states  constitute  what  may  be 
called  the  occurrent  or,  in  accordance  with  scholastic 
usage,  the  occasional  causal  factor.  The  occurrent  is 
distinguished  from  and  essentially  connected  with  the 
continuant  or  the  material  factor  in  causation.  The 
occurrent  and  continuant  factors  are  thus  united  in  our 
complete  conception  of  substance,  neither  being  con- 
ceivable apart  from  the  other.  This  analysis  gives 
meaning  to  the  conception  of  the  properties  of  the  con- 
tinuant, as  potential  causes  which  are  actualised  in 
accordance  with    unchanging  rules  by  the   relatively 


XXX  VUl 


INTRODUCTION 


INTRODUCTION 


XXXIX 


incidental  occurrences  that  come  into  being  either  from 
within  or  from  without  the  continuant.     In  the  former 
case  the  process  is  immanent,  cause  and  effect  being 
manifestations  of  the  changeless  nature  of  the  continuant, 
and  the  temporal  relation  between  cause  and  effect  is 
here  that  of  succession.    In  the  latter  case,  the  causality 
is  transeunt,   the    patient  being   that  whose  state  is 
determined,  the  agent  being  that  whose  alterable  relation 
to  the  agent  is  determinative.     Irt  t^ranseunt  causality 
the  temporal  relation  of  cause  to  effect  is  literal  simul- 
taneity, and   the   critical  instant  at  which  the  cause 
operates  is  that  in  which  there  is  also  literal  geometrical 
contact  of  cause  agent  with  effect  patient.     There  are 
two  fundamentally  distinct  types  of  transeunt  causality. 
i.*   In    the   one    case   no    change   of  state   in  the   agent 
accompanies  the  change  of  state  in  the  patient,  and  we 
X.  have  action  without  any  direct  reaction;  in  the  other 
^   case  change  of  state  in  the  one  directly  entails  change 
of  state  in  the  other  of  such  a  nature  that  the  latter  may 
be  formulated  as  a  function  of  the  former,  and  here 
action  always  involves  an  assignable  reaction.     The 
latter  case  holds  invariably  of  inter-physical  causality,  | 
and  again  of  inter-psychical  causality  within  the  sphere  < 
of  a  single  individual's  experience.     But  in  physico- 
psychical  causality,  as  also  in  psycho-physical  causality, 
action    never   directly  determines  reaction,   owing  to 
i  the  absolute  disparateness  between  the  physical  and 
psychical  in  regard  to  the  characters  of  the  states  which 
are  predicable  of  the  one  and  of  the  other.     It  is  here 
where  my  treatment  of  logical  questions  transgresses 
into  the  domain  of  ontology;  but  it  must  be  admitted 
that  all  logicians  who  treat  these  subjects  inevitably 
transgress  in  the  same  manner. 


(A)  The  position  assumed  by  probability  in  logical 
discussion  has  always  been  dubious.  On  the  one  side 
the  topic  has  been  assumed  to  be  the  exclusive  property 
of  the  mathematician,  or  rather  more  precisely,  the 
arithmetician.  On  this  view  the  quantity  called  prob- 
ability is  a  mere  abstract  fraction,  and  the  rules  of 
probability  are  merely  those  of  arithmetic.  The  fraction 
is,  in  short,  the  ratio  of  two  numbers,  the  number 
holding  for  a  species  to  that  holding  for  its  proximate 
genus,  this  ratio  being  necessarily  a  proper  fraction, 
the  limits  of  which  are  zero  and  unity.  If  this  view 
were  correct,  there  would  be  no  separate  topic  to  be 
called  probability.  A  precisely  reverse  account  of  prob- 
ability is  that  it  is  a  measure  of  a  certain  psychological  ^  i^u-ry^ 
attitude  of  thought  to  which  the  most  obvious  names 
that  could  be  given  are  belief  or  doubt,  taken  as  subject 
to  different  degrees.  On  either  of  these  two  extreme 
views  probability  would  have  no  particular  connection 
with  logic.  The  psychological  account  would  be  sepa- 
rated from  logic,  inasmuch  as  it  would  concern  solely 
the  causal  explanation  of  different  degrees  of  belief,  and 
would  thus  give  rise  to  no  principle  of  rational  criticism. 
The  mere  arithmetical  account  of  probability  ought  in 
the  first  instance  to  be  corrected  by  the  recognition  that 
the  topic  has  its  mental  side.  This  correction  requires 
that  probability  should  not  be  expressed  by  a  merely 
abstract  fraction,  but  rather  as  a  fraction  of  a  certain 
mental  quantity  which  may  be  called  certainly.  The 
psychological  conditions  of  the  variable  degrees  in  which 
doubt  may  approximate  to  certainty  are  as  such  outside 
the  province  of  logic;  but  when  these  various  degrees 
are  such  as  reason  would  dictate,  we  may  speak  of 
reasonable  doubt  as  an  assignable  fraction  of  certitude, 


J, 


icyit    iJ. 


a  INTRODUCTION 

thus  bringing  the  subject  into  the  sphere  of  logic. 
Further  the  quantity  or  degree  called  probability  at- 
taches exclusively  to  the  proposition ;  not  however  to 
the  proposition  as  such,  but  to  the  proposition  regarded 
as  based  upon  rationally  certified  knowledge  acquired 
by  any  supposed  thinker.  The  degree  of  probability  is 
therefore  referential  to  such  knowledge,  but  is  wholly 
independent  of  the  individual  thinker,  being  dependent 
solely  on  his  rational  nature,  and  the  knowledge  which 
he  has  rationally  acquired. 

The  whole  development  of  this  aspect  of  the  subject 
is  to  be  called  formal  probability,  and  constitutes  the 
one  subject  of  the  fourth  Part  of  this  work.    The  treat- 
ment of  probability  there  developed  must  be  distin- 
guished from  that  of  informal  probability,  that  is  required 
in  discussing  the  foundations  of  science  as  treated  in  my 
third  Part;  for  there,  while  the  logic  of  inductive  infer- 
ence is  made  to  depend  upon  the  principles  of  probability 
and  not  upon  any  big  fact  about  nature,  yet  probability 
is  only  introduced  on  broad  and  indeterminately  quan- 
titative lines.    This  treatment  leads  to  an  attempted 
enumeration   of   broadly   formulated   criteria   for   the 
evaluation  of  the  degrees  of  probability  to  be  attached 
to  the  generalisations  of  inductive  inference.    These 
criteria  are  merely  expressions  of  what  is  popularly 
felt,  and  their  rational  justification  can  only  be  repre- 
sented as  depending  upon  postulates :  that  is,  specula- 
tions that  are  neither  intuitively  self-evident  nor  ex- 
perientially  verifiable,  but  merely  demanded  by  reason 
in  order  to  supply  an  incentive  to  the  endeavour  to 
systematise  the  world  of  reality  and  thus  give  to  prac- 
tical action  an  adequate  prompting  motive. 


CHAPTER  I 


THE  PROPOSITION 


§  I.    A  SYSTEMATIC  treatment  of  logic  must  begin  by 
regarding  the  proposition  as  the  unit  from  which  the 
whole  body  of  logical  principles  may  be  developed.    A 
proposition  is  that  of  which  truth  and  falsity  can  be 
significantly  predicated.     Some  logicians  have   taken 
the  judgment  as  their  central    topic,   and  it  will    be 
necessary  to  examine  the  distinction  between  what  I 
have  called  a  proposition  and  what  appears  to  be  meant 
by  a  judgment.     It  has  been  very  generally  held  that  j 
the  proposition  is  the  verbal  expression  of  the  judgment; 
this,   however,   seems    to  be  an   error,    because  such 
characterisations  as  true  or  false  cannot  be  predicated 
of  a  mere  verbal  expression,   for  which   appropriate 
adjectives  would  be  'obscure,'  '  ungrammatical,'  'am- 
biguous,' etc.    There  appear  then  to  be  three  notions 
which,  though   intimately  connected,  must  be  clearly 
distinguished:  namely  (i)  what  may  be  called  the  sen- 
tence; (2)  the  proposition;  and  (3)  the  judgment.    The 
sentence  may  be  summarily  defined  as  the  verbal  ex- 
pression of  a  judgment  or  of  a  proposition;  it  remains, 
therefore,  to  distinguish  and  interrelate  the  proposition 
and  the  judgment. 

The  natural  use  of  the  term  judgment  is  to  denote 
an  act  or  attitude  or  process  which  may  constitute  an 
incident  in  the  mental  history  of  an  individual.  As  so 
conceived,  we  should  have  further  to  distinguish  the 

J.L.  I 


-i— 


■^^i^l-l 


M<4nr%  C'4' 


3  CHAPTER  I 

changing  phases  of  a  process  (which  might  alternately 
involve  interrogation,  doubt,  tentative  afifirmation  or 
negation)  from  the  terminus  of  such  process  in  which  a 
final  decision  replaces  the  variations  undergone  dunng 
what  is  commonly  called  suspense  of  judgment.     It 
would  thus  be  more  natural  to  speak  of  passing  judg- 
ment upon  a  proposition  proposed  in  thought  than  to 
identify  judgment  as  such  with  the  proposition.    This 
more  natural  usage  (which  is  that  which  I  shall  adopt) 
entails  the  necessity  of  recognising  the  distinction  be- 
tween various  attitudes  of  thought  on  the  one  hand, 
and  the  object  towards  which  that  thought  may  be 
directed  on  the  other ;  and  even  further,  when  necessary, 
of  recognising  the  adoption  of  any  of  these  alterable 
attitudes  of  thought  as  a  datable  occurrence  within  the 
total  experience  of  some  one  individual  thinker.    There 
will  thus  be  many  fundamental  attributes  that  must  be 
predicated  of  the  judgment  upon  a  proposition  different 
from,  and  often  .diametrically  opposed  to,  those  attributes 
that  are  to  be  predicated  of  the  proposition  itself. 

In  this  account  the  judgment  is  the  more  compre- 
hensive or  concrete  term,  since  when  seriously  treated 
it  involves  the  two  terms  thinker  and  proposition  and, 
in  addition,  the  occurrent  and  alterable  relation  that 
may  subsist  between  them.  In  thus  drawing  attention 
to  mental  process  in  my  exposition  of  logical  doctrine, 
I  am  taking  what  has  been  unfortunately  termed  a  '  sub- 
jective '  point  of  view.  For  the  term  '  subjective '  should 
be  substituted  'epistemic';  and  in  discarding  the  familiar 
antithesis  subjective  and  objective,  it  is  better  for  the  pur- 
poses of  Logic  to  substitute  the  antithesis  epistemic  and 
constitutive.  The  epistemic  side  of  logical  doctrine  points 

I.     I     I    .1     I    -I         V_|.    ,|M1-  ** 


THE  PROPOSITION 


3 


to  the  quite  universally  acknowledged  kinship  of  Logic 
with  Epistemology,  and,  in  using  this  term  in  preference 
to  subjective,  we  can  avoid  any  confusion  between  what 
belongs  to  Psychology  as  opposed  to  what  belongs  to 
Logic.  As  to  the  term  constitutive — a  term  for  which 
philosophers  are  indebted  to  Kant — it  has  the  force  of 
•  objective '  inasmuch  as  it  points  to  the  constitution  of 
such  an  object  of  thought-construction  as  the  proposition 
when  treated  independently  of  this  or  that  thinker.  I 
may  anticipate  what  will  be  treated  fully  in  the  later 
part  of  logical  doctrine,  by  pointing  out  that  the  dis- 
tinction and  connection  between  the  epistemic  and 
constitutive  sides  of  logical  problems  plays  an  important 
part  in  the  theory  of  Probability;  and,  in  my  view,  it 
ought  to  assume  the  same  importance  throughout  the 
whole  of  the  study  of  Logic. 

Now,  as  regards  the  relation  of  the  proposition  to 
any  such  act  as  may  be  called  judgment,  my  special 
contention  is  that  the  proposition  cannot  be  usefully 
defined  in  isolation,  but  only  in  connection  with  some 
such  attitude  or  act  of  thought ;  and  I  prefer  to  take  the 
notion  oi  asserting  as  central  amongst  these  variations  of 


%*»^. 


attitude— which  will  therefore  be  spoken  of  as  variations  "*^ 
in  the  assertive  attitude.  I  shall  also  maintain  that  the  ^  ^^ 
fundamental  adjectives  true  and  false  which  are  (perhaps 
universally)  predicated  of  mere  propositions  as  such, 
derive  their  significance  from  the  fact  that  the  proposition 
is  not  so  to  speak  a  self-subsistent  entity,  but  only  a 
factor  in  the  concrete  act  of  judgment.  Thus,  though 
we  may  predicate  of  a  certain  proposition — say  '  matter 
exists ' — that  it  is  true  or  that  it  is  false,  what  this  ulti- 
mately means  is,  that  any  and  every  thinker  who  might 

1—2 


0lf'wtm/i%0''  inJMpW%(#' 


^ 


4  CHAPTER  I 

at  any  time  assert  the  proposition   would  be   either 
exempt  or  not  exempt  from  error.    In  other  words,  the 
criticism  which  reason  may  offer  is  directed— not  to  the 
proposition— but  to  the  asserting  of  the  proposition; 
and  hence  the  customary  expression  that  such  and  such 
a  proposition   is   false   merely    means   that   anyones 
assertion  of  the  proposition  would  be  erroneous.    The 
1  equivalence  of  these  two  forms  of  criticism  follows  from 
the  fundamental  principle  that  an  attitude  of  assertion 
I  is  to  be  approved  or  condemned  in  total  independence 
'  of  the  person  asserting  or  of  the  time  of  his  assertion, 
I  and  in  exclusive  dependence  upon  the  content  of  his 
assertion.    This  fundamental  principle  of  Logic  will 
come  up  for  detailed  treatment  when  the  so-called  Laws 
of  Thought  are  explicitly  discussed.     In  order  to  mark 
the  important  distinction,  and  at  the  same  time  the  close 
connection,   between  the  proposition  and  the  act  of 
assertion,  I  propose  to  take  the  term  '  assertum '  as  a 
synonym  for  '  proposition'  when  such  terminology  may 
seem  convenient.    Thus,  the  assertum  will  coincide,  not 
exactly  with  what  has  been  asserted,  but  with  what  is 

in  its  nature  assertible. 

§  2.  Many  philosophers  have  used  the  term  belief 
in  its  various  phases  as  a  substitute  either  for  judgment 
or  for  assertion;  in  fact,  when  the  mental  aspect  of  any 
problem  assumes  special  prominence,  the  term  belief  as 
applied  to  the  proposition  is  more  naturally  suggested 
than  any  other.  While  the  object  of  belief  is  always  a 
proposition,  the  proposition  may  be  merely  entertained 
in  thought  for  future  consideration,  either  without 
being  believed,  or  in  a  more  or  less  specific  attitude 
opposed  to  belief,  such  as  disbelief  or  doubt.    To  doubt 


THE  PROPOSITION  5 

a  proposition  implies  that  we  neither  believe  nor  dis- 
believe it,  while  belief  and  disbelief  as  opposed  to  doubt 
have  in  common  the  mental  characteristic  of  assurance. 
Thus  there  are  three  opposed  attitudes  towards  a  pro- 
position, included  in  the  distinction  between  assurance 
and  doubt; — the  former  of  which  may  be  either  (assured) 
belief  or  (assured)  disbelief,  and  the  latter  of  which  ap- 
pears further  to  be  susceptible  of  varying  felt  degrees. 
The  close  association  amongst  all  the  terms  here  intro- 
duced brings  into  obvious  prominence  the  mental  side, 
which  such  terms  as  judgment  or  assertion  seem  hardly 
to  emphasise.    It  would  however,  I  think,  be  found  that 
there  is  in  reality  no  relevant  distinction  between  the 
implications  of  the  two  terms  *  judgment '  and  '  belief.' 
Those  logicians  who  have  spoken  exclusively  of  judg- 
ment, conception,   reasoning,   etc.,  have  had  in  view 
more  complicated  processes,  the  products  of  which  have 
been  explicitly  formulated ;  while  those  who  have  used 
belief  and  cognate  terms  have  included  more  primitive 
and  simple  processes,  the  products  of  which  may  not 
have  been  explicitly  formulated.     Since  the  traditional 
logic  has  treated  only  the  more  developed  processes, 
the  term  judgment  and  its  associates  is  perhaps  prefer- 
able for  this  somewhat  limited  view  of  the  scope  of 
Logic,  while  the  use  of  the  term  belief— which  must 
certainly  be  understood  to  include  the  higher  as  well  as 
the  lower  processes — points  to  a  wider  conception  of 
the  province  of  Logic.    To  put  the  matter  shortly,  I 
hold  it  to  be  of  fundamental  importance  to  insist  that 
there  is  some  factor  common  to  the  lower  and  higher 
stages,  and  that  this  common  factor,  to  which  the  name 
belief  has  been  given,  is  necessarily  directed  to  what 


6  CHAPTER  I 

in  Logic  is  called  a  proposition  \  Assertion,  in  the  sense 

here  adopted,  is  to  be  understood  to  involve  belief,  and 

may  be  defined  as  equivalent  to  conscious  belief.  This  de- 

j   finition  restricts  the  term  in  two  ways :  in  that,  firstly,  to 

..  assert  does  not  merely  mean  to  z^//^r  (without  belief);  and 

"^  secondly,  merely  to  believe  unconsciously  is  not  to  assert. 

§  3.    In  speaking  of  variations  of  attitude  towards 

the  proposition,  an  assumption  is  involved  that  there  is 

a  single  entity  called  the  proposition  that  is  the  same 

whatever   may   be   the   attitude   adopted   towards   it. 

Ordinary  language  supplies  us  with  names  for  such 

different  attitudes  along  with   cognate  names  for  the 

proposition :  thus  we  associate  '  to  assume '  with  '  an 

assumption  ' ;   'to  suppose '  with   '  a  supposition  * ;   *  to 

propose'  with  'a  proposition';  'to  postulate'  with  'a 

postulate ' ;   *  to  presume '  with  '  a  presumption  ' ;  etc. ' 

Consider  the  two  verbs  *to  assume'  and  *  to  presume.' 

It  will  be  acknowledged  that  these  denote  attitudes 

between  which  some  subtle  distinction  may  be  under- 

^  Readers  of  Psychology  should  be  warned  that,  when  psycho- 
logists contrast  *  imagination '  with  *  belief,'  each  term  indicates  an 
attitude  to  a  proposition ;  while,  when  they  contrast  *  imagination '  with 
*  perception,'  the  processes  to  which  they  refer  do  not  involve  any 
attitude  towards  a  proposition.  There  is  no  common  element  of 
meaning  in  these  two  applications  of  the  word  'imagination.' 

^  In  further  illustration  of  this  point  we  may  select  certain 
prominent  logical  terms  such  as  hypothesis,  postulate,  axiom.  Each 
of  these  terms  indicates  the  peculiar  attitude  to  be  assumed  towards 
the  proposition  in  question  by  any  thinker :  thus  a  hypothesis  stands 
for  a  proposition  which  awaits  further  scientific  investigation  before 
being  finally  accepted  or  rejected ;  a  postulate  stands  for  a  proposition 
which  cannot  be  brought  to  the  test  of  experience,  but  the  truth  of 
which  is  demanded  by  the  thinker;  and  an  axiom  is  a  proposition  the 
truth  of  which  is  self-evident  to  the  thinker. 


THE  PROPOSITION 


9 


stood;  and  thus  it  might  appear  that  in  correspond- 
ence with  this  distinction  there  must  be  a  similar  subtle 
distinction  between  an  assumption  and  a  presump- 
tion. Unfortunately  substantival  words  such  as  these 
are  apt  to  suggest  a  difference  in  nature  between  that 
which  in  the  one  case  is  presumed  and  in  the  other 
assumed ;  but  this  suggestion  must  be  rejected,  and  it 
must  be  maintained  on  the  contrary  that  the  content  of 
a  proposition  preserves  its  identity  unmodified,  inde- 
pendently of  all  variations  of  assertive  attitude  and  of 
personal  and  temporal  reference.  This  independence 
holds  also  in  regard  to  what  has  been  termed  *  logical ' 
in  contrast  with  *  psychological '  assertion.  The  phrase 
logically  asserted,  applied  to  this  or  that  proposition,  is 
only  metaphorically  legitimate,  and  literally  equivalent 
to  *  asserted  on  purely  rational  grounds  by  any  or  all 
rational  persons.'  In  other  words,  the  predicate  'as- 
serted '  conveys  no  meaning  when  taken  apart  from  a 
person  asserting. 

Adopting  as  we  do  the  general  view  that  no  logical 
treatment  is  finally  sound  which  does  not  take  account 
of  the  mental  attitude  in  thought,  it  follows  that  the 
fundamental  terms  *  true  '  and  *  false '  can  only  derive 
their  meaning  from  the  point  of  view  of  criticising  a 
certain  possible  mental  attitude.  We  are  thus  bound  to 
distinguish  the  object  of  this  attitude  (the  assertum) 
from  the  attitude  itself  which  may  vary  independently 
of  the  object ;  but  we  can  only  avoid  contradiction  or 
vagueness  if,  while  permitting  ourselves  to  distinguish 
between  the  attitude  and  its  object,  we  at  the  same 
time  refuse  to  separate  them.  We  may  further  explain 
the  adjectives  '  true  '  and  *  false  *  so  as  to  bring  out  what 


I 


8 


CHAPTER  I 


characterises  logic  in  contrast  with — or  rather  in  its 
relation  to — psychology :  namely  that  logic  formulates 
standards  or  imperatives  which  as  such  have  no  sig- 
nificance except  as  imposed  upon  mental  acts.  Thus 
we  may  say  that  the  application  of  the  adjectives  true 
and  false  coincides  with  the  application  of  the  imperatives 
'  to  be  accepted '  and  '  to  be  rejected '  respectively.  We 
may  add  that  these  imperatives  are  imposed  by  the 
thinker — in  the  exercise  of  his  reason — upon  himself. 
In  maintaining  this  coincidence  between  the  two  im- 
peratives on  the  one  hand  and  the  two  adjectives  (true 
and  false)  on  the  other,  it  must  not  be  taken  that  we 
are  able  thus  to  cle/ine  the  adjectives  true  and  false.  On 
the  contrary,  we  are  forced  to  insist  that  they  are  in- 
definable. We  are  only  indicating  that  a  reference  to 
mental  attitude  is  presupposed  when  Logic  recognises 
the  distinction  between  true  and  false  in  its  formulation 
of  standards  for  testing  the  correctness  of  a  judgment 
or  assertion. 

§  4.  So  far  we  have  taken  the  proposition  as  a  unit 
of  which  the  adjectives  true  and  false  may  be  predicated. 
Before  proceeding  to  analyse  the  proposition  into  its 
component  parts,  a  word  must  be  said  in  regard  to  the 
relation  of  logic  to  universal  grammar,  and  in  particular 
the  relation  between  grammatical  and  logical  analysis. 
Properly  speaking,  grammatical  analysis  cannot  be  re- 
garded as  dealing  merely  with  words  and  their  combi- 
nations. The  understanding  of  the  grammatical  structure 
of  a  sentence — which  includes  such  relations  as  those  of 
subject  to  predicate,  and  of  subordinate  to  co-ordinate 
clauses — requires  us  to  penetrate  below  the  mere  verbal 
construction  and  to  consider  the  formal  structure  of 


THE  PROPOSITION  9 

thought.  Hence,  on  the  one  hand,  grammar  cannot  be 
intelligently  studied  unless  it  is  treated  as  a  department  i 
of  logic ;  and,  on  the  other  hand,  logic  cannot  proceed 
without  such  a  preliminary  account  of  linguistic  structure 
as  is  commonly  relegated  to  grammar.  In  short,  uni-. 
versal  grammar  (as  it  is  called)  must  be  subsumed  under 
Logic.  On  this  view,  a  slight  alteration  in  grammatical 
nomenclature  will  be  required,  whereby,  for  the  usual 
names  of  the  parts  of  speech,  we  substitute  substantive- 
word  or  substantive-phrase,  adjective-word  or  adjective- 
phrase,  preposition- word  or  phrase,  etc.,  reserving  the 
terms  substantive,  adjective,  preposition,  etc.,  for  the 
different  kinds  of  entity  to  which  the  several  parts  of 
speech  correspond. 

§  5.  To  turn  now  to  the  analysis  of  the  proposition. 
We  find  that  in  every  proposition  we  are  determining  in 
thought  the  character  of  an  object  presented  to  thought 
to  be  thus  determined.  In  the  most  fundamental  sense, 
then,  we  may  speak  of  a  determinandum  and  a  deter- 
minans :  the  determinandum  is  defined  as  what  is  pre- 
sented to  be  determined  or  characterised  by  thought  or  I ! 
cognition ;  the  determinans  as  what  does  characterise  ' 
or  determine  in  thought  that  which  is  given  to  be  de- 
termined. We  shall  regard  the  substantive  (used  in  its 
widest  grammatical  sense)  as  the  determinandum,  and 
the  adjective  as  the  determinans.  Neither  of  these  terms 
can  be  defined  except  in  their  relation  to  one  another 
as  each  functions  in  a  possible  proposition.  As  it  has  ^ 
frequently  been  said,  the  proposition  is  par  excellence 
the  unit  of  thought.  This  dictum  means  that  the  logical 
nature  of  any  components  into  which  we  may  analyse 
the  proposition  can  only  be  defined  by  the  mode  in 


iA4»^**-» 


JL  •  jC  Jti^lw  » 


(W  /  •   it 


10 


CHAPTER  I 


THE  PROPOSITION 


II 


V*«Ab 


.i  which  they  enter  into  relation  within  it.  For  example, 
when  I  use  determinandum  for  the  substantive  and 
determinans  for  the  adjective,  I  am  only  defining  the 
one  in  terms  of  the  other,  inasmuch  as  the  common 
factor  '  determine '  is  contained  in  both.  This  account 
goes  beyond  that  which  has  become  commonplace  among 
many  philosophers,  namely,  that  the  subject  of  a  pro- 
position is  ultimately  something  which  cannot  be  defined 
in  the  way  in  which  a  predicate  or  adjective  can  be 
defined ;  for  to  this  we  have  to  add  that  the  predicate  of 
a  proposition  is  ultimately  something  which  cannot  be 
defined  in  the  way  in  which  a  subject  or  substantive  can 
be  defined.  These  two  statements  present  the  natures 
of  subject  and  predicate  purely  negatively,  the  positive 
element  being  supplied  by  the  terms  '  determinans '  and 
*  determinandum.' 

We  have  now  to  examine  the  nature  of  the  connec- 
tion involved  in  every  case  where  adjective  and  sub- 
stantive are  joined ;  for  example  *  a  cold  sensation,'  *  a 
tall  man.'  In  order  to  understand  the  verbal  juxta- 
^^7i^  i.^)  position  of  substantive  and  adjective,  we  must  recognise 
a  latent  element  of  form  in  thiis  cojistruct,  which  differen- 
tiates it  from  other  constructs— which  also  are  necessarily 
expressed  by  a  juxtaposition  of  words.  This  element 
of  form  constitutes  what  I  shall  call  the  characterising 
tie.  The  general  term  *  tie '  is  used  to  denote  what  is 
not  a  component  of  a  construct,  but  is  involved  in  under- 
standing the  specific  form  of  unity  that  gives  significance 
to  the  construct ;  and  the  specific  term  *  characterising 
tie'  denotes  what  is  involved  in  understanding  the 
junction  of  substantive  with  adjective.  The  invariable 
verbal  expression  for  the  characterising  tie  is  the  verb 


..*.—•*♦»». 


; 


'  to  be '  in  one  or  other  of  its  different  modes.  To  think 
of  '  a  tall  man '  or  of  '  a  cold  sensation '  is  to  think  of 
'  a  man  as  being  tall/  '  a  sensation  as  being  cold.'  Here 
the  word  *  being '  expresses  the  characterising  tie,  and 
the  fact  that  in  some  cases  the  word  may  be  omitted  is 
further  evidence  that  the  tie  is  not  an  additional  com-, 
ponent  in  the  construct,  but  a  mere  formal  element, 
indicating  the  connection  of  substantive  to  adjective. 
This  is  its  peculiar  and  sole  function ;  and,  as  the  ex- 
pression of  the  unique  connection  that  subsists  between 
substantive  and  adjective,  it  is  entirely  unmodifiable. 

The  distinction  and  connection  between  substantive 
and  adjective  correspond  to — and,  in  my  view,  explain 
— the  distinction  and  connection  between  particular  and 
universal\i  Ultimately  a  universal  means  an  adjective 
that  may  characterise  a  particular,  and  a  particular  means 
a  substantive  that  may  be  characterised  by  a  universal. 
The  terms  particular  (or  substantive)  and  universal  (or 
adjective)  cannot  be  defined  as  functioning  in  isolation, 
but  only  as  they  enter  into  union  with  one  another. 
There  is  some  danger  of  confusing  two  different  uses 
of  the  verb  *to  characterise,'  which  may  be  partly  re- 
sponsible for  the  historical  dispute  concerning  the  relation 
of  particular  to  universal,  w  Primarily  the  term  '  charac- 
terise' should  be  used  to  connect  substantive  with 
adjective  in  the  form  '  such  and  such  a  quality  or  ad- 
jective characterises  such  and  such  an  object  or  sub- 
stantive.' On  the  other  hand,  in  the  phrase  *  the  thinker 
characterises  such  or  such  an  object, 'characterises  means  I 

'  Here  the  terms  particular  and  universal  are  used  in  the  sense 
current  in  philosophy,  and  not  in  their  familiar  application  m 
elementary  logic,  where  they  stand  for  sub-divisions  of  the  proposition. 


J 


/C 


4fc  #'±lL:ii^ 


\f^^i^^   «^^^21      t 


r'iLAV*  •» 


i«^t><b»V^ 


C-^^-^^^XCi^tyr     ? 


Jl 


12 


CHAPTER  I 


THE  PROPOSITION 


13 


*cognitively  determines  the  character  of.'  Owing  to 
this  elliptical  use  of  the  term,  the  particular  has  been 
conceived  of  as  '  an  uncharacterised  object,'  and  this 
would  mean  literally  '  an  object  without  any  character  ' ; 
but  since  actually  every  object  must  have  character, 
the  only  proper  meaning  for  the  phrase  '  uncharacterised 
object '  is  *  an  object  whose  character  has  not  been 
cognitively  determined.'  If  then  the  term  '  exist '  may 
be  predicated  equally  of  a  universal  as  of  a  particular, 

I  then  we  may  agree  with  the  Aristotelian  dictum  that 
the  universal  exists,  not  apart  from,  but  m  tjhe  particular ; 
and  by  this  is  meant  that  the  adjective  exists,  not  apart 
from,  but  as  characterising  its  substantive;  to  which 
must  be  added  that  the  substantive  exists,  not  apart 
from,  but  as  characterised  by  its  adjective.  Now  in 
thought  the  substantive  and  the  adjective  may  be  said 
to  be  separately  and  independently  represented ;  hence 
thinking  effects  a  severance  between  the  adjective  and 
the  substantive,  these  being  reunited  in  the  asserted 
proposition — not  only  by  the  characterising  tie,  but  also 

»  by  what  we  may  call  the  assertive  tie.  The  blending 
of  the  assertive  with  the  characterising  tie  is  expressed 
in  language  by  the  transition  from  the  participial,  sub- 
ordinate, or  relative  clause,  to  the  finite  or  declaratory 
form  of  the  principal  verb.  Thus  in  passing  from  *  a 
child  fearing  a  dog '  to  '  a  child  fears  a  dog,'  the  charac- 
terising tie  joins  the  same  elements,  in  the  same  way, 
in  both  cases;  but  is,  in  the  latter,  blended  with  the 
assertive  tie.  That  the  ties  are  thus  blended  is  further 
shown  by  the  modifications  '  is-not,'  *  may  be,'  *  must  be,' 
by  which  the  verb  *  to  be '  is  inflected  in  order  to  indicate 
variations  in  the  assertive  attitude  while  the  character- 


ising relation  remains  unchanged.  The  copula  '  is '  of 
traditional  logic  is  thus  seen  to  be  a  blend  of  the  charac- 
terising with  the  assertive  tie. 

§  6.  We  must  now  criticise  a  view,  explicitly  opposed 
to  our  own,  as  to  the  nature  of  the  copula  is^  There 
has  been  for  a  long  period  an  assumption  that  the  pro- 
position in  some  way  or  other  asserts  the  relation  of 
identity.  This  relation  of  identity,  it  is  admitted,  is  not 
one  of  complete  or  absolute  identity,  but  involves  also 
a  relation  of  difference :  thus  the  proposition  '  Socrates 
is  mortal '  is  transformed  into  '  Socrates  is  a  mortal 
being' — where  'Socrates'  and  *a  mortal  being'  are 
affirmed  to  be  identical  in  denotation  but  different  in 
connotation.  Have  logicians  quite  recognised  the  ex- 
treme elaborateness  of  this  verbal  transformation  }  The 
adjective  *  mortal '  has  first  to  be  turned  into  a  sub-  t 
stantive  in  using  the  word  *  a  mortal  being ' ;  secondly,  i, 
the  indefinite  article  has  to  be  introduced,  since  it  is 
clear  that  Socrates  is  not  identical  with  every  mortal; 
thirdly,  the  indefinite  article  has  to  be  carefully  defined  3. 
as  meaning  one  or  other\  fourthly,  the  relation  of  the  ^ 
adjective  *  mortal '  to  the  substantive  *  being '  which  it 
characterises  still  remains  to  be  elucidated;  fifthly,  r 
another  adjective(a  relational  adjective)  n'dmely  identical 
is  introduced  in  the  compound  phrase  '  is  identical  with.' 
The  proposition  finally  becomes :  *  Socrates  is  identical 
with  one  or  other  being  that  is  mortal.'  Here  the  two 
adjectives  *  mortal '  and  *  identical  with  '  are  each  intro- 
duced after  is.  Now,  if  'is  identical  with'  is  to  be 
substituted  for  is  in  each  case,  then  we  shall  arrive  at 
an  infinite  regress.  Thus,  in  the  first  place,  *  Socrates 
is  identical  with  X '  (say)  must  be  rendered  '  Socrates 


'WCu^a. 


M 


CHAPTER  I 


THE  PROPOSITION 


« 


a 


is  identical  with  a  being  that  is  identical  with  X '  where 
the  force  of  is  still  remains  unexplained.  And  in  the 
second  place  '  one  or  other  being  that  is  mortal '  must 
be  rendered  *  one  or  other  being  that  is  identical  with 
a  mortal;  where  again  is  still  remains  to  be  explained. 
In  each  case,  if  an  infinite  regress  is  to  be  avoided,  the 
word  is  that  remains  must  be  interpreted  as  representing 
the  unique  mode  in  which  the  fundamentally  distinct 
categories  substantive  and  adjective  are  joined. 

§  7.  Having  so  far  considered  the  proposition  in  its 
mental  or  subjective  aspect,  we  have  next  to  examine  it 
in  what  may  be  called  its  objective  aspect.  Whereas  a 
proposition  is' related  subjectively  to  assertion,  we  shall 
find  that  it  is  related  objectively  \.ofact\  Our  conclusion, 
briefly  expressed,  is  that  any  proposition  characterises 
some  fact,  so  that  the  relation  of  proposition  to  fact  is 
the  same  as  that  of  adjective  to  substantive.  Bradley 
has  represented  a  proposition  as  ultimately  an  adjective 
characterising  Reality,  and  Bosanquet  as  an  adjective 
characterising  that  fragment  of  Reality  with  which  we 
are  in  immediate  contact.  In  adopting  the  principle 
that  a  proposition  may  be  said,  in  general,  to  characterise 
a  fact,  I  am  including  with  some  modification  what  is 
common  to  these  two  points  of  view. 

One  parallel  that  can  be  drawn  between  the  relation 
of  an  adjective  to  a  substantive  and  that  of  a  proposition 
to  a  fact  is  that,  corresponding  to  a  single  given  sub- 
stantive, there  are  an  indefinite  number  of  adjectives 
which  are  truly  predicable  of  it,  just  as  there  are  many 
different  propositions  which  truly  characterise  any  given 

1  Otherwise  expressed:  The  proposition,  subjectively  regarded,  is 
an  assertibik)  objectively  regarded,  dipossibile. 


fact.  Thus  we  do  not  say  that  corresponding  to  a  single 
fact  there  is  a  single  proposition,  but  on  the  contrary, 
corresponding  to  a  single  fact  there  is  an  indefinite 
number  of  distinct  propositions.  Again,  just  as  amongst  A 
the  adjectives  which  can  be  truly  predicated  as  charac- 
terising a  given  substantive,  some  are  related  to  others 
as  relatively  more  determinate ;  so,  amongst  the  several 
propositions  which  truly  characterise  a  single  fact,  some 
characterise  it  more  determinately  and  thus  imply  those 
which  characterise  the  same  fact;  less  determinately. 
We  may  therefore  regard  the  process  of  development 
in  thought  as  starting  from  a  fact  given  to  be  charac- 
terised, and  proceeding  from  a  less  to  a  continually 
more  determinate  characterisation. 

Again  there  is  an  exact  parallel  between  the  relation 
of  contradiction  or  contrariety  amongst  adjectives  that  -^ 
could  be  predicated  of  a  given  substantive,  and  amongst 
propositions  which  could  be  formulated  as  characterising 
a  given  fact.  Thus  the  impossibility  of  predicating 
certain  pairs  of  adjectives  of  the  same  substantive  in- 
volves the  same  principle  as  the  impossibility  of  charac- 
terising the  same  fact  by  certain  pairs  of  propositions: 
such  pairs  of  adjectives  and  propositions  ^r ^incompatible 
and  this  relation  of  incompatibility  lies  at  the  root  of 
the  notion  of  contradiction.  We  may  illustrate  the  re- 
lation of  incompatibility  amongst  adjectives  by  red  and 
green  regarded  as  characterising  the  same  patch.  It  is 
upon  this  relation  of  incompatibility  that  the  idea  of 
the  contradictory  not-red  depends;  for  not-red  means 
some  adjective  incompatible  with  red,  and  predicates 
indeterminately  what  is  predicated  determinately  by 
green,  or  by  blue,  or  hy yellow,  etc.  Amongst  propositions 


^  c-V^^  )  Aiv«-*i-^   <-  ^A4rp.,a.t.^  ->  ^*^  (  (ryvfc^) 


i6 


CHAPTER  I 


the  relation  of  incompatibility  may  be  illustrated  by 
'  Every  p  is  qu  and  '  Every  pv  is  non-^^,'  which  are 
more  determinate  forms  of  the  pair  of  contradictory 
propositions  '  Every  /  is  ^ '  and  '  Some  /  is  non-^.' 
These  latter  derive  their  significance  as  mutually  con- 
tradictory from  the  principle  that  the  actual  fact  must 
be  such  that  it  could  be  characterised  either  by  such  a 
relatively  determinate  proposition  as  '  Every  /is  qu 
or  by  such  a   relatively  determinate   proposition   as 

'  Every /t'  is  novt-q! 

This  account  of  the  relation  of  contradiction  as 
ultimately  derived  from  that  of  incompatibility  or  con- 
trariety  (whether  applied  to  adjectives  regarded  as 
characterising  substantives  or  to  propositions  regarded 
as  characterising  facts)  brings  out  in  another  aspect  the 
principle  that  any  given  substantive  or  any  given  fact 
may  be  truly  characterised  by  a  more  or  by  a  less 
determinate  adjective  or  proposition :  a  topic  which  will 
be  further  developed  in  later  chapters. 

The  above  logical  exposition  of  the  nature  of  a 
proposition  leads  to  a  consideration  of  the  philosophical 
problem  of  the  relation  of  thought  to  reality  in  one  of 
,  its  aspects.     It  is  at  the  present  day  agreed  that  this 
i  relation  cannot  be  taken  to  be  identity,  and  the  notion 
of  correspondence  has  been  put  forward  in  its  place. 
The  above  account  enables  us  to  give  a  more  definite 
exposition  of  what  more  precisely  this  so-called  corre- 
spondence entails :  the  truth  of  a  judgment  (expressed 
in  a  proposition)  may  be  said  to  mean  that  the  propo- 
sition is  in  accordance  with  a  certain  fact,  while  any 
proposition  whose  falsity  would  necessarily  follow  from 
the  truth  of  the  former  is  in  discordance  with  that  fact. 


i:HE  PROPOSITION  17 

In  this  way  the  somewhat  vague  conception  of  the 
correspondence  between  thought  and  reality  is  replaced 
by  the  relation  of  accordance  with  a  certain  fact  at- 
tributed to  the  true  proposition,  and  of  discordance  with 
the  same  fact  attributed  to  the  associated  false  propo- 
sition. 


J.  L. 


i8 


CHAPTER  II 

THE  PRIMITIVE  PROPOSITION 

§  I.    The  form  of  proposition  which  appears  to  be 
psychologically  prior  even  to  the  most  elementary  pro- 
position that  can  be  explicitly  analysed  is  the  exclama- 
tory  or  impersonal.    Propositions  of  this  kmd,  which 
are  more  or  less  unformulated  and  which  may  be  taken 
to  indicate  the  early  stages  in  a  developing  process,  will 
here  be  called  primitive.    The  most  formless  of  such 
primitive   propositions    is   the    exclamatory    assertion 
illustrated  by  such  an  utterance  as  ^  Lightning!      1  his 
appears  to  contain  only  a  characterising  adjective  with- 
out  any  assigned  subject  which  is  so  characterised.  Now 
it  is  true  that  any  proposition  can  be  regarded  as  a 
characterisation  of  the  universe  of  reality  regarded  as 
a  sort  of  unitary  whole ;  but  this  way  of  conceiving  the 
nature   of  the   proposition  in  general,   must  be  also 
associated  with  the  possibility  of  using  adjectives  as 
characterising  a  part  rather  than  merely  the  whole  of 
reality ;  and  certainly  the  case  here  is  one  in  which  we 
are  bound  to  recognise  the  lightning  as  having,  so  to 
speak,  an  assignable  place  within  the  universe,  and  not 
merely  as  an  adjective  attached  to  the  universe  as  a 
whole     The  lightning  as  an  actual  occurrence  must 
occupy  a  determinate  position,  in  reference  both  to  time 
and  to  space ;  but  it  is  obvious  that  no  reference  to  such 
determinate  position  is  itself  contained  in  the  merely 
exclamatory  assertion.    Any  implicit  reference  to  place 


THE  PRIMITIVE  PROPOSITION  19 

or  time  can  only  be  rendered  explicit  when  the  judg- 
ment has  been  further  developed ;  in  the  undeveloped 
judgment  the  reference  is  indeterminate,  and  any  judg- 
ment which  might  be  developed  from  this  primitive  form 
would  assert  what  v/as  unasserted  in  the  original.  In 
logical  analysis  it  is  of  the  utmost  importance  to  avoid 
putting  into  an  assertion  what  further  development  of 
the  percipient's  thought  might  elicit  on  the  basis  of  the 
original. 

We   ask   then,   how  such  judgment  in    its   most 
primitive  and  undeveloped  form  can  be  conceived  as 
referring  to  a  subject  when  its  verbal  expression  includes 
no  such  reference  ^    Now  we  may  speak  of  the  presented 
occasions  or  occurrences  that  give  rise   to  such  in- 
completely formulated  judgments  as  manifestations  of 
reality.    The  exclamatory  judgment  *  Lightning'  may 
thus  be  rendered  formally  complete  by  taking  as  subject 
term  'a  manifestation  of  reality. '   Here  I  do  not  propose 
to  take  simply  as  the  equivalent  of  the  exclamatory  judg- 
ment 'Reality  is  being  manifested  in  the  lightning,'  but 
rather  '  K  particular  portion  of  reality  manifests  the 
character  (indicated  by  the  adjectival   import  of  the 
word)  lightning.'     In  short,  what  is  asserted  by  the 
percipient  is  'a  manifestation  of  lightning\'  This  phrase 
for  representing  the  assertum  contains  of  course  the 
characterising  tie  but  not  the  assertive  tie.    The  asser- 
tive tie  may  be   introduced   by  employing  the  form: 
'There  is  a  manifestation  of  lightning,'  which  raises  the 
interesting  problem  as  to  the  significance  of  the  word 

^  In  grammatical  phraseology,  the  expression  'manifestation  of 
reality'  illustrates  the  subjective  genitive,  while  'manifestation  of 
lightning'  illustrates  the  objective  genitive. 

2—2 


20 


CHAPTER  II 


THE  PRIMITIVE  PROPOSITION 


'there.'    Like  many  other  words  in  current  language  it 
is  used  here  in  a  metaphorical,  or  perhaps  rather  in  a 
general  or  abstract  sense.    Literally  ^ there'  means  *m 
that  place/  so  that  in  its  original  significance  it  mvolves 
the  demonstrative  article,  and  furthermore— which  is 
the  new  matter  of  interest— a  reference  to  position  in 
space.    Moreover  the  tense  of  the  verb  is  points  to 
the  present  time.    If  these  references  were  developed 
still  more  precisely,  the  assertion  would  become:  '  There 
and  now— in  that  place  and  at  this  time— is  a  mam- 
festation  of  lightning.'    What  remains  as  the  significant 
element  in  the  word  ^ there  is,'  in  the  absence  of  any 
definite  reference  to  position  in  time  or  space,  must  be 
an  indefinite  reference  to  position  in  time  and  space. 
Otherwise  the  exclamatory  assertion  can  only  be  ex- 
pressed by  omitting  the  word  ^ there'  altogether,  and 
the   assertion   to   which  we   are  reduced— when   the 
subject  implicit  in  the  exclamation  is  made  explicit- 
becomes,  as  above,  'A  manifestation  of  lightning^' 

§  2.  The  phrase  *  there  is'  points  to  an  important 
presupposition  underlying  the  possibility  of  this  most 
primitive  form  of  perceptual  judgment:  namely,  that 
things  should  be  presented  apart  or  in  separation  in 
order  that  any  characterising  judgment  may  be  directed 
now  to  one  and  then  again  to  another.  Thus  separation 
'  of  presentment  is  a  presupposition  of  cognition  or  judg- 
ment. Here  I  use  the  word  *  presentment '  not  as 
equivalent  to  cognition,  but  as  something  presupposed 
in  all— even  the  most  primitive— acts  of  cognition.   The 

1  As  an  illustration  of  how  words  lose  their  philological  origin  and 
become  merely  metaphorical,  consider  the  expressions:  ^There  is  a 
God,'  *There  is  an  integer  between  5  and  7.' 


21 


word  '^^x^sent' — with  the  accent  on  the  second  syllable^ — 
is  in  English  equivalent  to  'give' ;  in  this  sense  2l pre- 
sentation is  eqivalent  to  a  datum — where  by  '  datum  '  is 
meant  not  a  piece  of  given  knowledge,  but  a  piece  of 
given  reality  that  is  to  be  characterised  in  knowledge. 
Thus  the  presentation  or  the  datum  is  what  I  have 
otherwise  called  the  determinandum — that  which  is 
given  or  presented  to  thought  to  be  thought  about.  This 
expresses  briefly,  the  meaning  of  the  primitive  '  this.' 
The  *  this '  as  thus  defined  is  not  rich  in  predicates  and 
adjectives,  but  at  the  same  time  it  cannot  be  said  to  be 
empty  of  adjectives  or  predicates,  because,  in  the  mean- 
ing of  thisness,  abstraction  is  made  from  all  predicates 
or  adjectives.  But  the  '  this '  cannot  be  explicated  apart 
from  an  implicit  reference  to  the  '  that,'  in  the  sense  that 
the  *this'  must  be  for  the  percipient  presented  in 
separation  from  the  *  that ' :  one  determinandum  is  one 
to  which  its  own  adjectives  may  be  assigned,  just  because 
the  other  must  be  presented  in  separation  or  apart  from 
the  one,  before  the  most  primitive  form  of  articulate 
judgment  is  possible^  Briefly  separateness  is  before  i 
relating;  more  specifically,  it  is  the  presupposition  which  • 
makes  it  possible  in  more  highly  developed  perception 
to  define  the  relations  (temporal  or  spatial)  between  those 
things  which  are  first  presented  merely  as  separate. 

^  When  accented  on  the  first  syllable,  its  meaning  combines  a 
reference  both  to  space  and  to  time;  so  that  the  word  presentation 
contains  in  its  meaning  the  three  factors  in  our  analysis,  viz.  the  given, 
the  here,  and  the  now. 

^  It  is  here  presumed  that  such  mental  processes  as  sense  differ- 
entiation, etc.,  in  which  the  experient  is  merely  passive  or  recipient, 
must  have  been  developed  prior  to  the  exercise  of  judgment,  to  furnish 
the  material  upon  which  the  activity  of  thought  can  operate. 


22 


CHAPTER  II 


THE  PRIMITIVE  PROPOSITION 


23 


It  is  in  this  quite  ultimate  sense  that  I  demur  to 
Mr  Bradley's  dictum  :  '  distinction  implies  difference/ 
His  dictum  means,  as  far  as  I  understand,  what  in  my 
own   terminology   I    should   express   by   the  phrase: 
'  otherness  presupposes  comparison '  (the  comparison,  in 
particular,  in  which  the  relation  of  difference  is  asserted). 
Now  in  my  view  this  dictum  is  exactly  wrong:  the 
assertion  of  'otherness'  does  not  presuppose  or  require 
a  previous  assertion  of  any  relation  of  agreement  or  of 
difference.    It  does  not  even  presuppose  the  possibility 
of  asserting  in  the  future  any  particular  relation  of 
agreement  or  of  difference.    The  first  important  relation 
which  will  be  elicited  from  otherness  is,  in  fact,  not  any 
relation  of  agreement  or  difference  at  all,  but  a  temporal 
or  spatial  relation ;  and  thus  the  primitive  assertion  of 
otherness  is  only  occasioned  and  rendered  possible  from 
the  fact  of  separateness  in  presentation.    When  presen- 
tations are  separate,  then  we  can  count  one,  two,  three; 
further,  we  can  connect  them  by  temporal  relations  such 
as  before  and  after,  or  by  spatial  relations  such  as  above 
or  below ;  and  finally  by  relations  of  comparison  such 
as  like  or  unlike.    These  examples  indicate  my  view  of 
^  the  quite  primary  nature  of  separateness  of  presentment, 
since  it  is  for  me  the  pre-requisite  for  all  those  acts  of 
[connecting  with  which  logic  or  philosophy — and   we 
!  may  add  psychology — is  throughout   concerned.     In 
illustration,  I  have  briefly  referred  only  to  relations  of 
number,  relations  of  time  and   space,  and   lastly  to 
relations  of  comparison  in  a  quite  general  sense. 
f   Summarising  this  attempt  to  indicate  the  precise 
logical  character  of  such  primitive  judgments  as  the  ex- 
clamatory or  impersonal,  and  their  relation  to  more 


highly  developed  judgments :  we  have  found  that  to 
assert  '  Lightning ! '  is  to  characterise,  not  reality  as  a 
whole,  but  a  separate  part  of  reality — to  use  an  inade- 
quate expression — and  that  the  possibility  for  this 
primitive  assertion  to  develop  into  higher,  more  inter- 
related forms  of  judgment,  is  wholly  dependent  on  the 
attribution  of  an  adjective  to  a  part  of  reality  presented 
in  separation  from  other  presentables. 

For  the  purpose  of  further  elucidation  we  may  bring 
two  or  three  assertions  into  connection  with  one  another, 
which  might  be  briefly  formulated  thus :  '  Lightning 
now ! '  '  Lightning  again !'  '  Thunder  then  ! '  The  first 
two  judgments  when  connected,  involve  two  manifesta- 
tions of  the  same  character  denominated  lightning, 
which  are  two  because  they  have  been  separately  pre- 
sented. The  use  of  the  terms  '  now  '  and  *  then '  does 
not  necessarily  presuppose  a  developed  system  of 
temporal  relations ;  but  they  indicate  at  least  the  possi- 
bility of  defining  relations  in  time  between  separately 
presented  manifestations.  Again  the  exclamation 
'Thunder!'  when  taken  in  connection  with  the  exclama- 
tion '  Lightning ! '  already  presupposes — not  only  that 
the  manifestations  are  given  somehow  in  separation — 
but  further  that  the  percipient  has  characterised  the 
separated  manifestations  by  different  adjectives.  I  will 
not  here  discuss  whether  these  manifestations  (as  I  have 
called  them)  are,  in  their  primitive  recognition,  merely 
the  individual's  sense-experiences  of  sound  and  light, 
or  whether  from  first  to  last  they  are  something  other 
than  sense-experiences.  In  either  case  our  logical  point 
will  be  the  same,  when  it  is  agreed  that  they  are  given 
separately,  and  that  their  separate  presentment  is  the 


24 


CHAPTER  II 


THE  PRIMITIVE  PROPOSITION 


25 


precondition  for  any  further  development  of  thought  or 
of  perception.  The  view  put  forward  here  is  so  far 
equivalent  to  Kant's  in  that  I  regard  space  and  time 
as  the  conditions  of  the  otherness  of  sense-experiences 
upon  which  the  possibility  of  cognising  determinate 
spatial  and  temporal  relations  depends,  and  that  this 
characteristic  of  space  and  time  is  what  constitutes 
Sense-Experience  into  a  manifold,  i.e.  a  plurality  of 
experiences,  which  we  can  proceed  to  count  as  many 
only  because  of  their  separate  presentment. 

Taking  more  elaborate  examples  of  these  primitive 
forms  of  perceptual  judgment:    *  This  is  a  flash    of 
lightning,'  '  This  (same)  flash  of  lightning  is  brighter 
than  that  (other),'  '  This  (same)  flash  appeared  before 
that  clap  of  thunder ' ;  we  note  that  in  the  predesigna- 
tions  *this'  and  'that'  the  percipient  has  passed  beyond 
the  indefinite  article  'a,'  and  has  identified  a  certain 
manifestation  as  that  of  which  more  than  one  characteri- 
sation can  be  predicated — e.g.  *  lightning'  and  'brighter 
than  that.'    It  is  this  identification  which  gives  to  the 
i  articles  '  this  '  and  '  that '  a  significance  which  may  be 
called  referential,  to  be  distinguished  from  their  use  as 
demonstratives;    and  in  this  alternation  between  the 
demonstrative  and  the  referential  usage,  we  can  trace, 
I  think,  the  very  primitive  way  in  which  thought  de- 
velops: first,  in  fixing  attention  upon  a  phenomenon 
by  pointing  to  its  position ;  and,  next,  in  identifying  it 
as  the  same  in  character  when  it  is  changing  its  spatial 
relations.   All  that  is  theoretically  required  for  identifica- 
tion is  the  retention — or  rather  the  detention — of  our 
cognition  or  judgment  upon  a  certain  manifestation;  but, 
when  attributing  different  qualities  or  relations  to  what 


continues  to  function  as  the  same  logical  subject,  we  are 
assisted  by  the  temporal  continuance  of  a  phenomenon, 
either  with  unaltered  quality,  or  in  an  unaltered  position, 
or  in  a  continuously  changing  position,  etc.  Thus  the 
changes  involving  variations  in  space,  time,  and  quality 
amongst  different  manifestations  of  the  same  pheno- 
menon constitute  the  groundwork  upon  which  the  several 
judgments  of  relation  are  built. 

In  asserting  '  There  was  a  flash  of  lightning  that 
was  very  brilliant  and  that  preceded  a  clap  of  thunder ' 
we  are  grasping  the  identity  of  a  certain  manifestation, 
thus  used  in  two  propositions,  of  which  one  predicates 
a  relation  in  time  to  a  clap  of  thunder,  and  the  other, 
a  quality  characterising  the  flash  itself.  Any  such  con- 
nected judgment  contains  implicitly  the  relation  of 
identity,  in  that  the  manifestation  is  maintained  as  an 
object  in  thought,  while  we  form  two  judgments  with 
' respect  to  it.  It  is  only,  in  short,  in  the  act  of  joining 
two  different  characterisations  that  any  meaning  for 
identity  can  be  found.  In  an  elementary  judgment 
which  predicates  only  one  adjective,  no  scope  or  signifi- 
cance for  the  notion  of  identifying  a  subject  as  such  can 
be  afforded.  Thus  the  three  factors  in  the  thinking 
process  which  the  '  this '  reveals  are  :  ( i )  the  ^iyen — 
which  is  equivalent  to  the  '  it '  in  *  It  lightens ! '  (2)  the 
demonstrative — which,  by  indicating  spatial  position, 
helps  towards  unique  identification,  (3)  the  referential — 
which  marks  the  achievement  of  this  process  of  identifi- 
cation. As  will  be  seen  from  the  discussion  in  a  subse- 
quent chapter,  these  three  elements  of  significance  in  the 
*this'  bring  it  into  line  with  the  proper  name. 


26 


COMPOUND  PROPOSITIONS 


27 


X   CHAPTER  III 

COMPOUND  PROPOSITIONS 

§  I.  Having  examined  the  proposition  in  its  more 
philosophical  aspects,  and  in  particular  from  the  point 
of  view  of  its  analysis,  the  present  chapter  will  be  mainly 
devoted  to  a  strictly  formal  account  of  the  proposition, 
and  will  be  entirely  concerned  with  the  synthesis  of 
propositions  considered  apart  from  their  analysis.  The 
chapter  is  intended  to  supply  a  general  introduction  to 
the  fundamental  principles  of  Formal  Logic ;  and  for- 
mulae  will  first  be  laid  down  without  any  attempt  at 
criticism  or  justification— which  will  be  reserved  for 
subsequent  discussion.  For  this  purpose  we  begin  by 
considering  the  different  ways  in  which  a  new  proposi- 
tion may  be  constructed  out  of  one  or  more  given 

propositions. 

In  the  first  place,  given  2.  single  proposition,  we  may 
construct  its  negative— expressed  by  the  prefix  not— 
not  -/  being  taken  as  equivalent  to/-false.  N  ext,  we  con- 
sider the  construction  of  a  proposition  out  of  two  or  more 
given  propositions.    The  proposition  thus  constructed 
will  be  called  compound,  and  the  component  proposi- 
tions out  of  which  it  is  constructed,  may  be  called  simple, 
relatively  to  the  compound,  although  they  need  not  be 
in  any  absolute  sense  simple.    The  prefix  not  may  be 
attached,  not  only  to  any  simple  proposition,  but  also 
to  a  compound  proposition,  any  of  whose  components 
again  may  be  negative. 


The  different  forms  that  may  be  assumed  by  com- 
pound propositions  are  indicated  by  different  conjunc- 
tions, A  proposition  in  whose  construction  the  only 
formal  elements  involved  are  negation  and  the  logical 
conjunctions  is  called  a  Conjunctional  Function  of  its 
component  propositions.  The  term  conjunctional  func- 
tion must  be  understood  to  include  functions  in  which 
negation  ^r  any  one  or  more  of  the  logical  conjunctions 
is  absent.  We  have  to  point  out  that  the  compound 
proposition  is  to  be  regarded,  not  as  a  mere  plurality  of 
propositions,  but  as  a  single  proposition,  of  which  truth 
or  falsity  can  be  significantly  predicated  irrespectively 
of  the  truth  or  falsity  of  any  of  its  several  components. 
Furthermore,  the  meaning  of  each  of  the  component 
propositions  must  be  understood  to  be  assignable  irre- 
spectively of  the  compound  into  which  it  enters,  so  that 
the  meaning  which  it  is  understood  to  convey  when 
considered  in  isolation  is  unaffected  by  the  mode  in 
which  it  is  combined  with  other  propositions. 

§  2.  We  will  proceed  to  enumerate  the  several  modes 
of  logical  conjunction  by  which  a  compound  proposition 
may  be  constructed  out  of  two  component  propositions, 
say/  and  q.  Of  all  such  modes  of  conjunction,  the  most 
fundamental  is  that  expressed  by  the  word  and:  this 
mode  will  be  called  par  excellence  conjunctive,  and  the 
components  thus  joined  will  be  called  conjuncts.  Thus 
the  compound  propositions — 

(a)  ''p  and  q^    {b)  '/  and  noX-q,'     {c)  *not-/  and  ^,'    {d)  *not-/  and  not-^,' 

are  the  conjunctive  functions  of  the  conjuncts/,  q-,  /, 
not-^ ;  not-/,  q ;  not-/,  not-^ ;  respectively.  There  are  thus 
four  distinct  conjunctive  forms  of  proposition  involving 
the  two  propositions/,  q,  taken  positively  or  negatively. 


28 


CHAPTER  III 


COMPOUND  PROPOSITIONS 


29 


The  significance  of  the  conjunctional  and  will  be 
best  understood  in  the  first  instance,  by  contrasting  it 
with  the  enumerative  and.    For  example,  we  use  the 
merely  enumerative  and  when  we  speak  of  constructing 
any  compound  proposition  out  of  the  components/ a/^^^. 
Here  we  are  not  specifying  any  mode  in  which/  and  q 
are  to  be  combined  so  as  to  constitute  one  form  of  unity 
rather  than  another;  we  are  treating  the  components 
(so  to  speak)  severally,  not  combinatorially.     In  other 
words,  the  enumeration^^  and  q — yields  two  propo- 
sitions, the  enumeration^;^  and  q  and  r — yields  three 
propositions,  etc.;  but  the  conjunctive  */  and  q,'  or  the 
conjunctive  'p  and  q  and  r  etc.,  yields  one  proposition. 
Again,  of  the  enumerated  propositions—/  and  q  and  r 
and... — some  may  be  true  and  others  false;   but  the 
conjunctive  proposition  */  and  q  and  r  and... '  must  be 
either  definitively  true  or  definitively  false.    Thus  in  con- 
joining two  or  more  propositions  we  are  realising,  not 
merely  the  force  of  each  considered  separately,  but  their 
joint  force.    The  difference  is  conclusively  proved  from 
the  consideration  that  we  may  infer  from  the  conjunc- 
tive proposition  'p  and  ^'  a  set  of  propositions  none 
of  which  could  be  inferred  from  /  alone  or  q  alone. 
The  same  holds,  of  course,  where  three  or  more  con- 
juncts  are  involved:  thus,  with  /,  q,  r,  as  components,^ 
seven  distinct  groups^  of  propositions  are  generated : 
viz.  the  three  groups  consisting  of  propositions  implied 
by  /,  by  q,  by  r  respectively ;  the  three  groups  con- 
sisting of  propositions  implied  by  'p  and  q'  by  */  and  r,' 
by  'q  and  r   respectively;  and  lastly,  the  group  con- 
sisting of  propositions  implied  by  /  and  q  and  r! 
^  The  term  group  is  here  used  in  its  precise  mathematical  significance. 


§  3.  In  our  first  presentation  of  formal  principles 
we  shall  introduce  certain  familiarly  understood  notions, 
such  as  equivalence,  inference  etc.,  without  any  attempt 
at  showing  how  some  of  them  might  be  defined  in  terms 
of  others.  The  same  plan  will  be  adopted  in  regard  to  the 
question  of  the  demonstrability  of  the  formal  principles 
themselves;  these  will  be  put  forward  as  familiarly  ac- 
ceptable, without  any  attempt  at  showing  how  some  of 
them  might  be  proved  by  means  of  others.  Ultimately, 
certain  notions  must  be  taken  as  intelligible  without 
definition,  and  certain  propositions  must  be  taken  as 
assertible  without  demonstration.  All  other  notions 
(intrinsically  logical)  will  have  to  be  defined  as  de- 
pendent upon  those  that  have  been  put  forward  with- 
out definition ;  and  all  other  propositions  (intrinsically 
logical)  will  have  to  be  demonstrated 2,^  dependent  upon 
those  that  have  been  put  forward  without  demonstration. 
But  we  shall  not,  in  our  first  outline,  raise  the  question 
of  the  dependence  or  independence  of  the  notions  and 
propositions  laid  down. 

Thus  the  formal  law  which  holds  of  Negation  is 
called  the  Law  of  Double  Negation:  viz.  not-not-/=/. 

§  4.  We  now  lay  down  the  formal  laws  which  hold 
of  compound  propositions  constructed  by  means  of  the 
conjunction  and.    They  are  as  follows: 


I. 


2. 


3. 


Laws  of  Conjunctive  Propositions 

The  Reiterative  Law: 

/  and  /  =/. 
The  Commutative  Law: 

/  and  q'^q  and  /. 
The  Associative  Law: 
(/  and  q)  and  r^p  and  (^  and  r). 


V 


30 


CHAPTER  III 


COMPOUND  PROPOSITIONS 


31 


Here  the  notion  of  equivalence  (expressed  by  the  short- 
hand symbol  =)  is  taken  as  ultimate  and  therefore  as  not 
requiring  to  be  defined.  These  laws  and  similar  formal 
principles  are  apt  to  be  condemned  as  trivial  Their 
significance  will  be  best  appreciated  by  reverting  to  the 
distinction  between  the  mental  acts  of  assertion  and 
progression  in  thought  on  the  one  hand,  and  the  propo- 
sitions to  which  thought  is  directed  on  the  other.  Thus 
1  the  laws  above  formulated  indicate,  in  general,  equiva- 
lence as  regards  the  propositions  asserted,  in  spite  of 
variations  in  the  modes  in  which  they  come  before 
thought.  Thus  the  content  of  what  is  asserted  is  not 
affected,  firstly,  by  any  r^-assertion ;  nor,  secondly,  by 
any  different  order  amongst  assertions ;  nor,  thirdly,  by 
any  different  grouping  of  the  assertions. 

§  5.  Having  considered  the  Conjunctive  form  of 
proposition,  we  turn  next  to  the  consideration  of  the 
remaining  fundamental  conjunctional  forms.  These  will 
be  classed  under  the  one  head  Composite  for  reasons 
which  will  be  apparent  later.  So  far,  compound  propo- 
sitions have  been  divided  into  the  two  species  Con- 
junctive and  Composite,  and  we  shall  now  proceed  to 
subdivide  the  latter  into  four  sub-species,  each  of  which 
has  its  appropriate  conjunctional  expression,  viz. : 


Lu^*>M^t*w    X^)   '^^^  Direct-Implicative  function  of/,  q  : —     \ip  then  q.- 
^(2)   The  Counter-Implicative  function  of/,  q\ —  \i  q  then/.* 

((3)   The  Disjunctive  function  of/,  q : —     Not-both/  and  q* 


j^Hj^ 


C«'*~*»**-'>'«-**<»<^J^ 


•(4)   The  Alternative  function  of/,  q : — 


Either/  or  q. 


In  the  implicative  function  '\{p  then  Oy  p  is  implicans^ 
and  g  implicate ;  in  the  counter-implicative  function  *  If  ^ 
then/,'/ is  implicate  and  q  is  implicans^;  in  the  dis- 

*  The  plural  of  implicans  must  be  written :  implicants. 


junctive  function  '  Not-both  /  and  q'  p  and  q  are  dis- 
juncts;  and  in  the  alternative  function  *  Either/  ox  q, 
p  and  q  are  alternants.  These  four  functions  of  /,  q, 
are  distinct  and  independent  of  one  another.  The 
technical  names  that  have  been  chosen  are  obviously  in 
accordance  with  ordinary  linguistic  usage.  The  impli- 
cative and  counter-implicative  functions  are  said  to  be 
Complementary  to  one  another,  as  also  the  disjunctive 
and  alternative  functions.  Each  of  the  four  other  pairs, 
viz.  (i)  and  (3);  (i)  and  (4);  (2)  and  (3);  (2)  and  (4) 
may  be  called  a  pair  of  Supplementary  propositions. 
These  names  are  conveniently  retained  in  the  memory 
by  help  of 

The  Square  of  Independence 


^ 


•  v»      supplementary 


,C^vv«4/»'^^H/'C-^'   "p*" 


Now  when  we  bring  into  antithesis  the  four  con- 
junctive functions : 

(i)  /  and  not-$r ;     (2)  not-/  and  q\      (3)  /  and  q ;      (4)  not-/  and  not-i^; 

with  the  four  composite  functions: 

(i)  if/ then  ^;   (2)  if-then/;   (3)  not-both  /  and  ^ ;   (4)  either/ or  ^;     Cr^ri^!^^  f^:^:^ 

we  shall  find  that  each  of  the  composite  propositions  is 
equivalent  to  the  negation  of  the  corresponding  con- 
junctive. This  is  directly  seen  in  the  case  (3)  of  the 
conjunctive  and  the  disjunctive  functions  of/,  q.    Thus, 


32 


CHAPTER  III 


COMPOUND  PROPOSITIONS 


33 


'Not-both/  and  q'  is  the  direct  negative  of  'Both 
/  and  q!  Again,  for  case  (4)  *  Either  /  or  q'  is  the 
obvious  negative  of  'Neither  /  nor  q'  The  relations 
of  negation  for  all  cases  may  be  derived  by  first 
systematically  tabulating  the  equivalences  which  hold 
amongst  the  composite  functions,  as  below :  abbreviating 
not-/  and  not-^  into  the  forms  p  and  q  respectively. 

Table  of  Equivalences  of  the  Composite  Functions 

Counter- 
Implicative         Implicative  Disjunctive  Alternative 

Form  Form  Form  Form 

1.  If/  then  q=\i  q  then  ^  =  Not  both  p  and  ^  =  Either  p  ox  q 

2.  If/  then  q  =  \i  q  then  /  =  Not  both  p  and  $r  =  Either  p  ox  q 

3.  If/  then  ^=If  ^  then  /  =  Not  both  p  and  ^  =  Either  p  ox  q 

4.  If  p  then  q^li  q  then  /  =  Not  both  /  and  ^  =  Either  /  or  ^ 

In  the  above  table  it  will  be  observed: 

{a)  That  each  composite  function  can  be  expressed 
in  four  equivalent  forms :  thus,  any  two  propositions  in 
the  same  row  are  equivalent,  while  any  two  propositions 
in  different  rows  are  distinct  and  independent. 

{b)  That  the  propositions  represented  along  the 
principal  diagonal  are  expressed  in  terms  of  the  positive 
components/,  q]  being  in  fact  identical  respectively 
with  the  implicative,  the  counter-implicative,  the  dis- 
junctive and  the  alternative  functions  of/,  q. 

{c)  That  all  the  remaining  propositions  are  ex- 
pressed as  functions  of  /  and  not-^,  or  of  not-/  and  q, 
or  of  not-/  and  not-^. 

We  may  translate  the  equivalences  tabulated  above 
in  the  form  of  equivalences  of  functions,  thus : 

I.  The  implicative  function  of/,  q]  the  counter- 
implicative  function  of  not-/,  noX.'q]  the  disjunctive  func- 


tion of/,  not-^;  and  the  alternative  function  of  not-/,  q, 
are  all  equivalent  to  one  another.    Again, 

2.  The  counter-implicative  function  of/,  q ;  the  im- 
plicative function  of  not-/,  not-^;  the  disjunctive  function 
of  not-/,  q\  and  the  alternative  function  of/,  not-^,  are 
all  equivalent  to  one  another.     Again, 

3.  The  disjunctive  function  of/,  q;  the  alternative 
function  of  not-/,  not-q\  the  implicative  function  of  /, 
not-^;  and  the  counter-implicative  function  of  not-/,  q, 
are  all  equivalent  to  one  another.    Again, 

4.  The  alternative  function  of/,  q]  the  disjunctive 
function  of  not-/,  not-^;  the  implicative  function  of 
not-/,  ^;  and  the  counter-implicative  function  of/,  not-f, 
are  all  equivalent  to  one  another. 

Since  the  force  of  each  of  the  four  composite  func- 
tions of/,  q  can  be  represented  by  using  either  the 
Implicative  or  the  Counter-implicative  or  the  Disjunc- 
tive or  the  Alternative  form,  the  classification  of  the 
four  functions  under  one  head  Composite  is  justified. 
And  since  each  Composite  function  is  equivalent  to  a 
certain  Disjunctive  proposition,  it  is  also  equivalent  to 
the  negation  of  the  corresponding  Conjunctive  propo- 
sition.   Thus : 

1.  The  implicative  '  lip  then  q'  negates  the  Conjunctive  '/  and  q.' 

2.  The  counter-   1  ,.^      , 

implicative/  I^i^then/  „       „  „  ^/and^r.' 

3.  The  disjunctive  'Not  both/  and  q'         „       „  „  '/  and  qJ 

4.  The  alternative      'Either/   or  ^'         „       „  „  '/and^.' 

Thus  inasmuch  as  no  composite  function  is  equiva- 
lenl  to  any  conjunctive  function,  we  have  justified  our 
division  of  compound  propositions  into  the  two  funda- 
mentally opposed  species  Conjunctive  and  Composite. 

J.L.  ^ 


34 


CHAPTER  III 


The  distinction  and  relation  between  these  compo- 
site forms  of  proposition  may  be  further  brought  out  by 
tabulating  the  inferences  in  which  a  simple  conclusion 
is  drawn  from  the  conjunction  of  a  composite  with  a 
simple  premiss.  In  traditional  logic,  the  Latin  verbs 
ponere  (to  lay  down  or  assert)  and  tollere  (to  raise  up 
or  deny)  have  been  used  in  describing  these  different 
modes  of  argument.  The  gerundponendo  (by  affirming) 
or/^//^;/rf'(?  (by  denying)  indicates  the  natureofthe(simple) 
premiss  that  occurs ;  while  the  participle  ponens  (affirm- 
ing) or  tollens  (denying)  indicates  the  nature  of  the 
(simple)  conclusion  that  occurs :  the  validity  or  invalidity 
of  the  argument  depending  on  the  nature  of  the  com- 
posite premiss.  There  are  therefore  four  modes  to  be 
considered  corresponding  to  the  four  varieties  of  the 
composite  proposition,  thus : 


Table  of  Valid  Modes 

Modus 

1.  Ponendo  Ponens  :  If/ then ^;  but/ 

2.  ToUendo  Tollens  :  If  q  then/ ;  but  / 

3.  Ponendo  Tollens  :  Not  both/  and  q ;  but/ 

4.  Tollendo  Ponens  :      Either/   or  ^;but/ 


9 


Form  of 
Composite  Premiss 

The  Implicative. 
The  Counter-Implic. 
The  Disjunctive. 
The  Alternative. 


The  customary  fallacies  in  inferences  of  this  type 
I  may  be  exhibited  as  due  to  the  confusion  between  a 
composite  proposition  and  its  complementary: 


Table  of  Invalid  Modes 


Modus 

1.  Ponendo  Ponens  :  If  q  then/ ;  but  / 

2.  Tollendo  Tollens  :  If/  then  q ;  but  / 

3.  Ponendo  Tollens  :      Either/    or   ^;but/ 

4.  Tollendo  Ponens  :  Not  both/  and  q\  but/ 


9 


Form  of 
Composite  Premiss 

The  Counter-ImpHc. 
The  Implicative. 
The  Alternative. 
The  Disjunctive. 


COMPOUND  PROPOSITIONS  35 

The  rules  for  correct  inference  from  the  above  table 
of  valid  modes  may  be  thus  stated : 

1.  From  an  implicative,  combined  with  the  affirma- 
tion of  its  implicans,  we  may  infer  the  affirmation  of  its 
implicate. 

2.  From  an  implicative,  combined  with  the  denial 
of  its  implicate,  we  may  infer  the  denial  of  its  implicans. 

3.  From  a  disjunctive,  combined  with  the  affirma- 
tion of  one  of  its  disjuncts,  we  may  infer  the  denial  of 
the  other  disjunct. 

4.  From  an  alternative,  combined  with  the  denial 
of  one  of  its  alternants,  we  may  infer  the  affirmation  of 
the  other  alternant. 

§   6.    We  ought  here  to  refer  to  an  historic  con- 
troversy as  regards  the  interpretation  of  the  conjunction 
*or.'    It  has  been  held  by  one  party  of  logicians  that  i 
what  I  have  called  the  Alternative  form  of  proposition, 
viz.,  that  expressed  by  either-or,  should  be  interpreted 
so  as  to  include  what  I  have  called  the  Disjunctive,  viz., 
that  expressed  by  not-both.   This  view  has  undoubtedly 
been  (perhaps  unwittingly)  fostered  by  the  almost  uni- 
I  versal  misemployment  of  the  term  Disjunctive  to  stand 
for  what  ought  to  be  called  Alternative.    This  prevalent 
confusion  in  terminology  has  led  to  a  real  blunder  com- 
mitted by  logicians.     The  blunder  consists  in  the  falla- 
cious use  of  the  Ponendo  Tollens  as  exhibited  in  the  table 
above  given.   Consider  the  argument :  'A  will  be  either 
first  or  second';   'It  is  found  that  A  is  second';  there- 
fore A  is  not  first'  Here  the  conclusion  is  represented  as 
:  following  from  the  promising  qualifications  of  the  candi- 
j  date  A,  whereas  it  really  follows  from  the  premiss  'A  can- 
■not  be  both  first  and  second.'    In  fact,  the  Alternative 

3-2 


36 


CHAPTER  III 


COMPOUND  PROPOSITIONS 


37 


#--41.-/' 


proposition  which  is  put  as  premiss  is  absolutely  irrele- 
vant to  the  conclusion,  which  would  be  equally  correctly 
inferred  whether  the  alternative  predication  were  false 
or  true. 

It  remains  then  to  consider  whether  the  logician 
can  properly  impose  the  one  interpretation  of  the  alter- 
native form  of  proposition  rather  than  the  other.  The 
reply  here,  as  in  other  similar  cases,  is  that,  in  the 
matter  of  verbal  interpretation,  the  logician  can  impose 
legislation — not  upon  others — but  only  upon  himself. 
However,  where  any  form  of  verbal  expression  is  ad- 
mittedly ambiguous,  it  is  better  to  adopt  the  interpreta- 
tion which  gives  the  smaller  rather  than  the  greater  force 
to  a  form  of  proposition,  since  otherwise  there  is  danger 
of  attaching  to  the  judgment  an  item  of  significance 
beyond  that  intended  by  the  asserter.  This  principle  of 
interpretation  has  the  further  advantage  that  it  compels 
the  speaker  when  necessary  to  state  unmistakeably  and 
explicitly  what  may  have  been  implicitly  and  perhaps 
confusedly  present  in  his  mind.  I  have  therefore  adopted 
as  my  interpretation  of  the  form  Either-or  that  smaller 
import  according  to  which  it  does  not  include  NoUboth. 
Those  logicians  who  have  insisted  on  what  is  called  the 
^A^W-  'exclusive'  interpretation  of  the  alternative  form  of  pro- 
position (i.e.  the  interpretation  according  to  which  ^^M^r- 
or  includes  Not-botK)  seem  sometimes  to  have  been 
guilty  of  a  confusion  between  what  a  proposition  asserts, 
and  what  may  happen  to  be  known  independently  of 
the  proposition.  Thus  it  may  very  well  be  the  case 
that  the  alternants  in  an  alternative  proposition  are 
almost  always  'exclusive'  to  one  another;  but  this,  so 
far  from  proving  that  the  alternative  proposition  affirms 


this  exclusiveness,  rather  suggests  that  the  exclusive- 
ness  is  a  fact  commonly  known  independently  of  the 
special  information  supplied  by  the  alternative  proposi- 
tion itself. 

In  this  connection,  the  significance  of  the  term  com- 
plementary which  I  have  applied  to  the  implicative  and 
counter-implicative  as  well  as  to  the  disjunctive  and 
alternative,  may  be  brought  out.  Propositions  are  ap- 
propriately called  complementary  when  a  special  im-  >c 
portance  attaches  to  their  conjoint  assertion  \  Thus  it 
may  be  regarded  as  an  ideal  of  science  to  establish  a 
pair  of  propositions  in  which  the  implicans  of  the  one 
is  the  implicate  of  the  other ;  and  again  to  establish  a 
number  of  propositions  which  are  mutually  co-disjunct 
and  collectively  co-alternate.  The  term  complementary 
is  especially  applicable  where  propositions  are  conjoined 
in  either  of  these  ways,  because  separately  the  propo- 
sitions represent  the  fact  partially,  and  taken  together 
they  represent  the  same  fact  with  relative  completeness. 

We  next  consider  the  inferences  that  can  be  drawn 
from  the  conjunction  of  two  supplementary  propositions. 
These  may  be  tabulated  in  two  forms,  the  first  of  which 
brings  out  the  fundamental  notion  of  the  Dilemma ;         [>iUfruruL 
and  the  second  that  of  the  Reductio  ad  Im^possibile, 

First  Table  for  the  Conjunction  of  Supplementaries 

The  Dilemma 
(i)  *  If  >  then  $^' and   (4)  *  If/ then  ^':  therefore,^. 
(3)  *If/  then  q'  and   (2)  *If/  then  q' \  therefore,  q. 

(2)  *  If  ^  then  /'  and   (4)  *  If  ^  then  p ' :  therefore,  p, 

(3)  *If$r  then/' and  (i)  *If^then/':  therefore,/. 

^  Thus  complementary  propositions  might  be  defined  as  those 
which  [are  frequently  confused  in  thought  and  frequently  conjoined       • 
in  fact. 


38 


CHAPTER  III 


f 


The  above  table  illustrates  the  following  principle : 
The  conjunction  of  two  implicatives,  containing  a 
common  implicate  but  contradictory  implicants,  yields 
the  affirmation  of  the  simple  proposition  standing  as 
common  implicate.    Or  otherwise : 

Any  proposed  proposition  must  be  true  when  its 
tmth  would  be  implied  both  by  the  supposition  of  the 
truth  and  by  the  supposition  of  the  falsity  of  some  other 
proposition. 

Second  Table  for  the  Conjunction  of  Supplementaries 
The  Redtictio  ad  Impossibile 

(i)  *If^  then/' and   (4)  'If-then/':  therefore,  $r. 
(3)  'If$^  then/' and   (2)  *If^then/':  therefore,^. 

(2)  *  If/ then  ^' and   (4)  *If/then^':  therefore,/. 

(3)  'If/ then  ^' and   (i)  *If/then$^':  therefore,/. 

This  second  table  illustrates  the  following  principle : 

The  conjunction  of  two  implicatives,  containing  a 
common  implicans  but  contradictory  implicates,  yields 
the  denial  of  the  simple  proposition  standing  as  com- 
mon implicans.    Or  otherwise  : 

Any  proposed  proposition  must  be  false  when  the 
supposition  of  its  truth  would  imply  (by  one  line  of 
argument)  the  truth  and  (by  another  line  of  argument) 
the  falsity  of  some  other  proposition. 

§  7.  In  tabulating  the  formulae  for  Composite  pro- 
positions as  above  I  have  merely  systematised  (with 
slight  extensions  and  modifications  of  terminology) 
what  has  been  long  taught  in  traditional  logic ;  and  it 
is  only  in  these  later  days  that  criticisms  have  been 
directed  against  the  traditional  formulae,  especially  on 
the  ground  that  their  uncritical  acceptance  has  been  found 


COMPOUND  PROPOSITIONS 


39 


to  lead  to  certain  paradoxical  consequences,  which  may 
be  called  the  Paradoxes  of  Implication.  In  this  con- 
nection it  is  (I  think)  desirable  to  explain  what  is  meant 
by  a  paradox.  When  a  thinker  accepts  step  by  step  Jarud^ 
the  principles  or  formulae  propounded  by  the  logician 
until  a  formula  is  reached  which  conflicts  with  his 
common-sense,  then  it  is  that  he  is  confronted  with  a 
paradox.  The  paradox  arises — not  from  a  merely  blind 
submission  to  the  authority  of  logic,  or  from  any  arbi- 
trary or  unusual  use  of  terms  on  the  logician  s  part — 
but  from  the  very  nature  of  the  case,  as  apprehended 
in  the  exercise  of  powers  of  reasoning  with  which  every- 
one is  endowed.  In  particular,  the  paradoxes  of  impli- 
cation are  not  due  to  any  unnatural  use  of  the  lerm 
implication,  nor  to  the  positing  o{  ^ny  fundamental  for- 
mula that  appears  otherwise  than  acceptable  to  com- 
mon sense.  It  is  the  formulae  that  are  derived — by 
apparently  unexceptionable  means  from  apparently  un- 
,  exceptionable  first  principles — that  appear  to  be  excep- 

■  tionable. 

Let  us  trace  the  steps  by  which  we  reach  a  typical 
paradox.  Consider  the  alternative  'Not-/  or  q!  If 
this  alternative  were  conjoined  with  the  assertion  */,' 
we  should  infer  'q!  Hence,  *  Not-/  or  ^'  is  equivalent 
to  '  If  /  then  q!  Similarly  'p  or  q'  is  equivalent  to  '  If 
not-/  then  q!  Now  it  is  obvious  that  the  less  deter- 
minate statement  'por  q'  could  always  be  inferred  from 
the  more  determinate  statement  '/':  e.g.  from  the  rela- 
tively determinate  statement  *^  is  a  solicitor'  we  could 
infer  'A  is  a  solicitor  or  a  barrister'  i.e.  'A  is  a  lawyer,' 
Hence,  whatever  proposition  ^  may  stand  for,  we  can 
infer  'p  ox  q   from  'f\  or  again,  whatever/  may  stand 


40 


CHAPTER  III 


COMPOUND  PROPOSITIONS 


41 


for  we  can  infer  'not-/  or  q'  from  q.  Hence  (i)  given 
*/'  we  may  infer  *If  not-/  then  ^/  and  (ii)  given  'q 
we  may  infer  *If  /  then  q,'  whatever  propositions  / 
and  q  may  stand  for.  These  two  consequences  of  the 
uncritical  acceptance  of  traditional  formulae  have  been 
expressed  thus:  (i)  A y^/?^  proposition  (e.g.  not-/ when 
/  has  been  asserted)  implies  any  proposition  (e.g.  ^) ; 
(ii)  A  /r/^^  proposition  (e.g.  q,  when  q  has  been  asserted) 
\  is  implied  by  any  proposition  (e.g.  /).  Thus  '2  +  3  =  7' 
»  would  imply  that  'It  will  rain  to-morrow';  and  *It  will 
rain  to-morrow*  would  imply  that  *  2  -h  3  =  5.'  That  these 
two  implicative  statements  are  technically  correct  is 
shown  by  translating  them  into  their  equivalent  alterna- 
tive forms,  viz.:  (i)  'Either  2  -f  3  is  unequal  to  7  or  it  will 
rain  to-morrow';  (ii)  'Either  it  will  not  rain  to-morrow 
or  2  +  3  =  5.'  W^  ^^y  certainly  say  that  one  or  other 
of  the  two  alternants  in  (i)  as  also  in  (ii)  is  true,  the 
other  being  of  course  doubtful. 

Taking  'If/  then  q'  to  stand  for  the  paradoxically 
reached  implicative  in  both  cases,  we  have  shown  that 
(i)  from  the  denial  of  /  (the  implicans),  and  (ii)  from 
the  affirmation  of  q  (the  implicate)  we  may  pass  to  the 
assertion  '  If/  then  q'    This  is,  of  course,  only  another 
way  of  saying  that  the  implicative   'If  /  then  q'  is 
equivalent  to  the  alternative  '/  false  or  q  true.'    Thus 
when  we  know  that  '  If/  then  q'  is  true,  it  follows  that 
.  we  know  that  'either/  is  false  or  q  is  true' ;  but  it  does 
'  not  follow  that  either  'we  know  that  /  is  false'  or  'we 
I  know  that  q  is  true.'    The  paradoxically  reached  im- 
plicative merely  brings  out  the  fact  that  this  may  be  so 
in  some  cases',  i.e.  when  asserting  'If/  then  q,'  there 
are  cases  in  which  we  know  that  '/  is  false,'  and  there 


I 


are  cases  in  which  we  know  that  'q  is  true.'  But  it  is 
proper  to  enquire  whether  in  actual  language — literary 
or  colloquial — the  implicative  form  of  proposition  is  ever 
introduced  in  this  paradoxical  manner.  On  the  one  hand, 
we  find  such  expressions  as:  'If  that  boy  comes  back, 
I'll  eat  my  head';  'If  you  jump  over  that  hedge,  I'll  give 
you  a  thousand  pounds';  'If  universal  peace  is  to  come 
tomorrow,  the  nature  of  mankind  must  be  very  different 
from  what  philosophers,  scientists  and  historians  have 
taken  it  to  be';  etc.,  etc.  Such  phrases  are  always  in- 
terpreted as  expressing  the  speakers  intention  to  deny 
the  implicans  \  the  reason  being  that  the  ^^ar^r  is  assumed 
to  be  ready  to  deny  the  implicate.  Again,  on  the  other 
hand,  we  find  such  forms  as:  'If  Shakespeare  knew  no 
Greek,  he  was  not  incdpable  of  creating  great  tragedies.' 
^If  Britain  is  a  tiny  island,  on  the  British  Empire  the 
sun  never  sets.'  *If  Boswell  was  a  fool,  he  wrote  a 
work  that  will  live  longer  than  that  of  many  a  wiser 
man.'  '  If  Lloyd  George  has  had  none  of  the  advantages 
of  a  public  school  education,  it  cannot  be  maintained 
that  he  is  an  unintelligent  politician.'  Such  phrases  are 
always  interpreted  as  expressing  the  speaker  s  intention 
to  affirm,  the  implicate,  the  reason  being  that  the  hearer 
}  may  be  assumed  to  be  willing  to  affirm  the  implicans. 

Looking  more  closely  into  the  matter  we  find  that 
when  a  speaker  adopts  the  implicative  form  to  express 
his  denial  of  the  implicans,  he  tacitly  expects  his  hearer 
to  supplement  his  statement  with  a  tollendo  tollens\  and 
when  he  adopts  it  to  express  his  affirmation  of  the 
implicate,  he  expects  the  hearer  to  supplement  it  with 
2.  ponendo  ponens.  Furthermore,  inasmuch  as  the  alter- 
native form  of  proposition  requires  to  be  supplemented 


42 


CHAPTER  ni 


COMPOUND  PROPOSITIONS 


43 


by  ^tollendo  [ponens)  and  the  disjunctive  by  ^Lponendo 
{tollens),  we  find  that  an  implicative  intended  to  ex- 
press the  denial  of  its  implicans  is  quite  naturally  ex- 
pressed otherwise  as  an  alternative:  e.g.  *That  boy 
won't  come  back  or  I'll  eat  my  head,'  to  which  the  hearer 
is  supposed  to  add  'But  you  won't  eat  your  head'; 
therefore  (I  am  to  believe  that)  ^the  boy  won't  come 
h^ck'  {tollendo ponens)\  and  we  find  that  an  implicative 
intended  to  express  the  affirmation  of  its  implicate  is 
quite  naturally  expressed  otherwise  as  a  disjunctive : 
e.g.  *It  cannot  be  held  that  Shakespeare  both  knew  no 
Greek  and  was  incapable  of  creating  great  tragedies,'  to 
which  the  hearer  is  supposed  to  add  *  But  Shakespeare 
knew  no  Greek,'  and  therefore  (I  am  to  believe  that)  ^he 
was  capable  of  creating  great  tragedies '  {ponendo  tollens). 
We  have  yet  to  explain  how  the  appearance  of 
paradox  is  to  be  removed  in  the  general  case  of  a  com- 
posite being  inferred  from  the  denial  of  an  implicans 
(or  disjunct)  or  from  the  affirmation  of  an  implicate 
(or  alternant).  Now  the  ordinary  purpose  to  which  an 
implicative  (or,  more  generally,  a  composite)  proposition 
is  put  is  inference :  so  much  so  that  most  persons  would 
hesitate  to  assert  the  relation  expressed  in  a  composite 
proposition  unless  they  were  prepared  to  use  it  for 
purposes  of  inference  in  one  or  other  of  the  four  modes, 
ponendo  ponens,  etc.  In  other  words,  Implication  is 
naturally  regarded  as  tantamount  to  Potential  Inference. 
Now  when  (i)  we  have  inferred  'Up  then  q'  from  the 
denial  of  */,'  can  we  proceed  from  'Up  then  q'  con- 
joined with  p'  to  infer  'q'}  In  this  case  we  join  the 
affirmation  of  '/'  with  a  premiss  which  has  been  in- 
ferred from  the  denial  of  / ;  and  this  involves  Contra- 


diction,  so  that  such  an  inference  is  impossible.  Again,  £. 
when  (ii)  we  have  inferred  'Up  then  q'  from  the  affirm- 
ation of  '^,'  can  we  proceed  from  'Up  then  q'  con- 
joined with  'p'  to  infer  'q'>  In  this  case  we  profess  to 
infer  *^'  by  means  of  a  premiss  which  was  itself  inferred 
from  'q'\  and  this  involves  Circularity,  so  that  this  fc 
inference  again  must  be  rejected.  The  solution  of  the  ,. 
paradox  is  therefore  found  in  the  consideration  that 
though  we  may  correctly  infer  an  implicative  from  the 
denial  of  its  implicans,  or  from  the  affirmation  of  its 
implicate,  or  a  disjunctive  from  the  denial  of  one  of  its 
disjuncts,  or  an  alternative  from  the  affirmation  of  one 
of  its  alternants,  yet  the  implicative,  disjunctive  or  al- 
ternative so  reached  cannot  be  applied  for  purposes  of 
further  inference  without  committing  the  logical  fallacy 
either  of  contradiction  or  of  circularity.  Now  it  must 
be  observed  that  the  rhetorical  or  colloquial  introduction 
of  a  paradoxical  composite,  which  is  meant  to  be  inter- 
preted as  the  simple  affirmation  or  denial  of  one  of  its 
components,  achieves  its  intention  by  introducing — as 
the  other  component  of  the  composite — a  proposition 
whose  falsity  or  truth  (as  the  case  may  be)  is  palpably 
obvious  to  the  hearer.  The  hearer  is  then  expected  to 
supplement  the  composite  by  joining  it  with  the  obvious 
affirmation  or  denial  of  the  added  component,  and 
thereby,  in  interpreting  the  intention  of  the  speaker,  to 
arrive  at  the  proposition  as  conclusion  which  the  speaker 
took  as  his  first  premiss.  Accordingly  the  process  of 
interpretation  consists  in  taking  the  same  propositions 
in  the  same  mode  and  arrangement  as  would  have  en- 
tailed circularity  if  adopted  by  the  speaker. 

§  8.     The  distinction  between  an  implicative  pro- 


44 


CHAPTER  III 


position  that  can  and  one  that  cannot  be  used  for  in- 
ferential purposes  may  now  be  further  elucidated  by 
reference  to  the  distinction  between  Hypothesis  and 
Assertion.  In  order  that  an  implicative  may  be  used 
for  inference,  both  the  implicans  and  the  implicate  must 
be  entertained  hypothetically.  In  the  case  o{ ponendo 
ponens  the  process  of  inference  consists  in  passing  to 
the  assertion  of  the  implicate  by  means  of  the  assertion 
of  the  implicans,  so  that  the  propositions  that  were 
entertained  hypothetically  in  the  implicative,  come  to 
be  adopted  assertively  in  the  process  of  inference.  The 
same  holds,  mutatis  mutandis,  for  the  other  modes. 
Now  when  we  have  inferred  an  implicative  from  the 
affirmation  of  its  implicate  or  from  the  denial  of  its 
implicans — as  in  the  case  of  the  implicative  which  ap- 
pears paradoxical — the  two  components  of  the  impli- 
cative thus  reached  cannot  both  be  regarded  as  having 
been  entertained  hypothetically;  and  hence  the  prin- 
ciple according  to  which  inference  is  a  process  of  pass- 
ing from  propositions  entertained  hypothetically  to  the 

.  same  propositions  taken  assertorically,  would  be  vio- 
lated if  we  used  the  composite  for  inference.  This  con- 
sideration constitutes  a  further  explanation  of  how  the 

S  paradoxes  in  question  are  solved. 

The  above  analysis  may  be  symbolically  represented 
by  placing  under  the  letter  standing  for  a  proposition 
the  sign  \-  to  stand  for  assertorically  adopted  and  the 
sign  H  for  hypothetically  entertained. 

Thus  the  fundamental  formula  for  correct  inference 
may  be  rendered : 

From  */  would  imply  q*  with/;  we  may  infer  q, 


COMPOUND  PROPOSITIONS  45 

where,  in  the  implicative  premiss,  both  implicans  and 
implicate  are  entertained  hypothetically. 

Now  the  following  inferences,  which  lead  to  para- 
doxical consequences,  may  be  considered  correct:  i.e. 

{a)   From  q,  we  may  infer  'p  would  imply  q: 

{b)    From/,  we  may  infer  '/  would  imply  g? 
V  h  H 

But  the  implicative  conclusions  here  reached  cannot 
be  used  for  further  inference:  i.e. 

{c)    From  'p  would  imply  q'  with/;  we  cannot  infer  <7. 
{d)  From  */  would  imply  q'  with/;  we  cannot  infer  0. 

For  in  {c)  the  implicate,  and  in  {d)  the  implicans 
enters  assertorically,  and  these  inferences  therefore  con- 
travene the  above  fundamental  formula  which  requires 
that  both  implicate  and  implicans  should  enter  hypo- 
thetically. Thus  while  admitting  {a)  that  'a  true  pro- 
position would  be  implied  by  any  proposition,'  yet  we 
cannot  admit  {c)  that  *a  true  proposition  can  be  inferred 
from  any  proposition.'  Similarly,  while  admitting  {b) 
that  *a  false  proposition  would  imply  any  proposition,' 
yet  we  cannot  admit  [d)  that  *from  a  false  proposition 
we  can  infer  any  proposition.'  In  fact,  the  attempted 
inference  {c),  where  the  conclusion  has  already  been  as- 
serted, would  entail  circularity ;  and  the  attempted  infer- 
ence {(£),  where  the  premiss  has  already  been  denied, 
would  involve  contradiction. 

Still  maintaining  the  equivalence  of  the  composite 
propositions  expressible  in  the  implicative,  the  counter- 
implicative,  the  alternative  or  the  disjunctive  form,  each 
of  these  four  forms  will  give  rise  to  a  like  paradox.    The 


46 


CHAPTER  III 


following  table  gives  all  the  cases  in  which  we  reach  a 
Paradoxical  Composite;  that  is,  a  Composite  which 
cannot  be  used  for  inference,  either  in  the  modus  po- 
nendo  ponens,  tollendo  tollens,  ponendo  tollens  or  tollendo 
ponens.  The  sign  of  assertion  in  each  composite  must 
be  interpreted  to  mean  asserted  to  be  true  when  the 
term  to  which  it  is  attached  agrees  with  the  premiss,  and 
asserted  to  be  false  when  it  contradicts  the  premiss. 

Table  of  Paradoxical  Composites 
{a)    From  q  we  may  properly  infer 

(r)  p  ox  q—\ip  then  q=\iq  then /  =  Not  both  i>  and  q. 
Hf-H[-|-H  Hh 

(2)  p  or  q  =  lip  then  q  =  l( q  then /  =  Not  both/  and  q. 


or 


(Jf)    From  q  we  may  properly  infer 

(3)  P  or  q  =  I{p  then  q  —  liq  then/  =  Not  both/  and  q^ 
or  (4)   /  or  ^  =  If/ then  ^=  If  ^  then  ^  =  Not  both/ and  ^. 

The  above  composites  can  never  be  used  for  further 
inference.    Thus : 

in  line  (i),  the  attempted  inference 

*'p  -''  q^  would  be  circular  and  'not-^  .*.  not-/'  would  be  contradictory ; 
in  line  (2),  the  attempted  inference 

*not-/  .*.  q^  would  be  circular,  and  *not-^  .*./'  would  be  contradictory; 

in  line  (3),  the  attempted  inference 

*not-/  .*.  not-^'  would  be  circular,  and  V  •*•/'  would  be  contradictory ; 
in  line  (4),  the  attempted  inference 

*/  .*.  not-$r'  would  be  circular,  and  ^q  .'.  not-/'  would  be  contradictory. 

The  paradox  of  implication  assumes  many  forms, 
some  of  which  are  not  easily  recognised  as  involving 
mere  varieties  of  the  same  fundamental  principle.     But 


COMPOUND  PROPOSITIONS  ^ 

I  believe  that  they  can  all  be  resolved  by  the  consider- 
ation that  we  cannot  without  qualification  apply  a  com- 
posite and  (in  particular)  an  implicative  proposition  to 
the  further  process  of  inference.  Such  application  is 
possible  only  when  the  composite  has  been  reached 
irrespectively  of  any  assertion  of  the  truth  or  falsity  of 
its  components.  In  other  words,  it  is  a  necessary  con- 
dition for  further  inference  that  the  components  of  a 
composite  should  really  have  been  entertained  hypo- 
thetically  when  asserting  that  composite. 

§  9.    The  theory  of  compound  propositions  leads  to 
a  special  development  when  in  the   conjunctives  the 

components  are  taken — not,  as  hitherto,  assertorically 

but  hypothetically  as  in  the  composites.  The  conjunc- 
tives will  now  be  naturally  expressed  by  such  words  as 
possible  or  compatible,  while  the  composite  forms  which 
respectively  contradict  the  conjunctives  will  be  expressed 
by  such  words  as  necessary  or  impossible.  If  we  select 
the  negative  form  for  these  conjunctives,  we  should  write 
as  contradictory  pairs : 


Conjunctives  {possible) 

a.  p  does  not  imply  q 

b.  p  is  not  implied  by  q 

c.  p  is  not  co-disjunct  to  q 

d.  P  is  not  co-alternate  to  q 


Composites  {necessary) 

a.  p  implies  q 

b.  p  is  implied  by  q 

c.  p  is  co-disjunct  to  q 

d.  p  is  co-alternate  to  q 


Or  Otherwise,  using  the  term  ^possible' throughout, 
the  four  conjunctives  will  assume  the  form  that  the  several 
conjunctions—/^,/^,  pq  and /^— are  respectively /t?^- 
sible.  Here  the  word  possible  is  equivalent  to  being 
merely  hypothetically  entertained,  so  that  the  several 
conjunctives  are  now  qualified  in  the  same  way  as  are 
the  simple  components  themselves.     Similarly  the  four 


48 


CHAPTER  III 


COMPOUND  PROPOSITIONS 


49 


corresponding  composites  may  be  expressed  negatively 
by  using  the  term  'impossible,'  and  will  assume  the 
form  that  the  ^^;y*unctions  pq^  pq,  pq  and  pq  are  re- 
spectively impossible,  or  (which  means  the  same)  that 
the  ^wjunctions/^, /^,  pq  a.nd pq  are  necessary.  Now 
just  as  'possible'  here  means  merely  * hypothetically 
entertained/  so  'impossible'  and  'necessary'  mean  re- 
spectively 'assertorically  denied'  and  *assertorically 
affirmed/ 

The  above  scheme  leads  to  the  consideration  of  the 
determinate  relations  that  could  subsist  of  /  to  ^  when 
these  eight  propositions  (conjunctives  and  composites) 
are  combined  in  everypossibleway  without  contradiction. 
Prima  facie  there  are  i6  such  combinations  obtained  by 
selecting  a  or  a,  b  or  b,  c  or  c,  d  or  ^for  one  of  the  four 
constituent  terms.  Out  of  these  i6  combinations,  how- 
ever, some  will  involve  a  conjunction  of  supplementaries 
(see  tables  on  pp.  2i1^  38),  which  would  entail  the  as- 
sertorical  affirmation  or  denial  of  one  of  the  components 
p  or  q,  and  consequently  would  not  exhibit  a  relation  of 
p  to  q.  The  combinations  that,  on  this  ground,  must  be 
disallowed  are  the  following  nine : 

died,  abed,  abed,  abed]  abed,  baed,  cabd,  dabc\  abed. 

The  combinations  that  remain  to  be  admitted  are 
therefore  the  following  seven : 

abed,  edab\  abed,  bald,  edab,  deal',  abed. 

In  fact,  under  the  imposed  restriction,  since  a  or  b 
cannot  be  conjoined  with  e  or  d,  it  follows  that  we  must 
always  conjoin  a  with  c  and  d\  b  with  c  and  d\  e  with 
a  and  l ;  d  with  a  and  ~b.    This  being  understood,  the 


■     seven  permissible  combinations  that  remain  are  properly 
to  be  expressed  in  the  more  simple  forms: 

ab,  cd',  ab,  ba,  cd,  dc\  and  abed 

These  will  be  represented  (but  re-arranged  for  purposes 
of  symmetry)  in  the  following  table  giving  all  the 
possible  relations  of  any  proposition  /  to  any  proposition 
q.  The  technical  names  which  1  propose  to  adopt  for 
the  several  relations  are  printed  in  the  second  column 
of  the  table. 

Table  of  possible  relations  of  propositio7i  p  to  proposition  q. 


1.  {a,  b) :  p        implies  and  is  implied  by        q. 

2.  {a,  b) :  p      implies  but  is  not  implied  by     q. 

3.  {b,a)\  p  is  implied  by  but  does  not  imply  q, 

4.  {a,  b,  c,(f):  p  is  neither  implicans  nor  impli 

cate  nor  co-disjunct  nor  co-alternate  to  q 

5.  (^,_?) 

6.  {c,d) 

7.  {c,d) 


:) 


/  is  co-implicant  to  q- 
p  is  super-implicant  to  q. 
p  is  sub-implicant  to  q. 

p    is  independent  of    q,  -J 

p  is  sub-opponent  to  q. 
p  is  super-opponent  to  q. 
i)    is  co-opponent  to    q. 


(fm%%/fj-t-n,^iAf-r^r^  y 


:  p  is  co-alternate  but  not  co-disjunct  to  q. 
:  pis  co-disjunct  but  not  co-alternateto^. 
:  p  is  co-disjunct  and  co-alternate  to  q. 

Here  the  symmetry  indicated  by  the  prefixes,  co-, 
super-,  sub-,  is  brought  out  by  reading  downwards  and 
upwards  to  the  middle  line  representing  independence. 
In  this  order  the  propositional  forms  range  from  the 
supreme  degree  of  consistency  to  the  supreme  degree 
of  opponency,  as  regards  the  relation  of/  to  ^.  In  tradi- 
tional logic  the  seven  forms  of  relation  are  known  respec- 
tively by  the  names  equipollent,  superaltern,  subaltern, 
independent,  sub-contrary,  contrary,  contradictory.  This 
latter  terminology,  however,  is  properly  used  to  express 
the  formal  relations  of  implication  and  opposition, 
whereas  the  terminology  which  I  have  adopted  will  apply 
indifferently  both  for  formal  and  for  material  relations. 


^ 


4 


J.L. 


48 


CHAPTER  III 


corresponding  composites  may  be  expressed  negatively 
by  using  the  term  *  impossible,'  and  will  assume  the 
form  that  the  ^^;^junctions  pq,  pq,  pq  and  pq  are  re- 
spectively impossible,  or  (which  means  the  same)  that 
the  ^^'ijunctions/^, /^,  pq  3.nd pq  are  necessary.  Now 
just  as  ^possible'  here  means  merely  * hypothetically 
entertained/  so  'impossible'  and  'necessary'  mean  re- 
spectively /assertorically  denied'  and  'assertorically 
affirmed/ 

The  above  scheme  leads  to  the  consideration  of  the 
determinate  relations  that  could  subsist  of  p  to  q  when 
these  eight  propositions  (conjunctives  and  composites) 
are  combined  in  everypossibleway  without  contradiction. 
Prima  facie  there  are  i6  such  combinations  obtained  by 
selecting  aora.b  or  3,  c  or  c,  d  or  ^for  one  of  the  four 
constituent  terms.  Out  of  these  i6  combinations,  how- 
ever, some  will  involve  a  conjunction  of  supplementaries 
(see  tables  on  pp.  37,  38),  which  would  entail  the  as- 
sertorical  affirmation  or  denial  of  one  of  the  components 
p  or  q,  and  consequently  would  not  exhibit  a  relation  of 
p  to  q.  The  combinations  that,  on  this  ground,  must  be 
disallowed  are  the  following  nine : 

cibcd,  aicdy  abed,  abcd\  abed,  bacd,  cabd,  dabc\  abed. 

The  combinations  that  remain  to  be  admitted  are 
therefore  the  following  seven : 

abed,  cdab\  abed,  bald,  edab,  dcab',  abed. 

In  fact,  under  the  imposed  restriction,  since  a  or  b 
cannot  be  conjoined  with  c  or  d,  it  follows  that  we  must 
always  conjoin  a  with  c  and  d\  b  with  c  and  d\  c  with 
a  and  b ;  d  with  a  and  ~b.    This  being  understood,  the 


COMPOUND  PROPOSITIONS  49 

seven  permissible  combinations  that  remain  are  properly 
to  be  expressed  in  the  more  simple  forms: 

ab,  cd\  ab,  ba,  cd,  dc;  and  abed. 

These  will  be  represented  (but  re-arranged  for  purposes 
of  symmetry)  in  the  following  table  giving  all  the 
possible  relations  of  any  proposition  /  to  any  proposition 
q.  The  technical  names  which  1  propose  to  adopt  for 
the  several  relations  are  printed  in  the  second  column 
of  the  table. 

Table  of  possible  relations  of  proposition  p  to  proposition  q. 


1.  {a,b):  p        implies  and  is  implied  by        q. 

2.  {a,  b) :  p     implies  but  is  not  implied  by     g. 

3.  (b,-n) :  p  is  implied  by  but  does  not  imply  q. 

4.  (a,  b,  c^d):  p  is  neither  implicans  nor  impli 

cate  nor  co-disjunct  nor  co-alternate  to  q 

5.  {d,c) 

6.  {c,d) 

7.  {c,d) 


;) 


/  is  co-implicant  to  q- 
p  is  super-implicant  to  q. 
p  is  sub-implicant  to   q. 


p    is  independent  of    q.  -^ 

p  is  sub-opponent  to   q. 
p  is  super-opponent  to  q. 


:  p  is  co-alternate  but  not  co-disjunct  to  q. 
:  pis  co-disjunct  but  not  co-alternateto^. 
:  /  is  co-disjunct  and  co-alternate  to  ^.    \  p    is  co-opponent  to    q. 

Here  the  symmetry  indicated  by  the  prefixes,  co-, 
super-,  sub-,  is  brought  out  by  reading  downwards  and 
upwards  to  the  middle  line  representing  independence. 
In  this  order  the  propositional  forms  range  from  the 
supreme  degree  of  consistency  to  the  supreme  degree 
of  opponency,  as  regards  the  relation  of/  to  ^.  In  tradi- 
tional logic  the  seven  forms  of  relation  are  known  respec- 
tively by  the  names  equipollent,  superaltern,  subaltern, 
independent, sub-contrary, contrary,  contradictory.  This 
latter  terminology,  however,  is  properly  used  to  express 
the  formal  relations  of  implication  and  opposition, 
whereas  the  terminology  which  I  have  adopted  will  apply 
indifferently  both  for  formal  and  for  material  relations. 


#Ci4lii*iM^^ 


dh^^,  'k         fit  fllB. 


J.  L. 


4   • 


50 


CHAPTER  IV 

SECONDARY  PROPOSITIONS  AND  MODALITY 

§  I.    The  division  of  propositions  into  simple  and 
compound  is  to  be  distinguished  from  another  division 
to  which  we  shall  now  turn,  namely  that  into  primary 
and  secondary.    A  secondary  proposition  is  one  which 
predicates  some  characteristic  of  a  primary  proposition. 
While  it  is  unnecessary  to  give  a  separate  definition  of 
a  primary  proposition,  a  tertiary  proposition  may  be  de- 
fined as  one  which  predicates  a  certain  characteristic  of  a 
secondary  proposition,  just  as  a  secondary  proposition 
predicates  some  characteristic  of  a  primary  proposition. 
Theoretically  this  succession  of  propositions  of  higher 
and  higher  order  could  be  carried  on  indefinitely.    But 
it  should  be  observed  that  any  adjective  that  can  be 
predicated  of  2.  primary  proposition  can  be  significantly 
predicated  of  a  proposition  as  such,  i.e.  equally  of  a 
primary,  a  secondary,  and  a  tertiary,  etc.  proposition  ; 
and  that,  in  consequence,  although  propositions  may  be 
ranged  into  higher  and  higher  orders,  adjectives  pre- 
dicate of  propositions  are  of  only  one  order,  and  will 
be  called  *'pre-propositional."    Taking  /  to  stand  for 
any  proposition  we  may  construct  such  secondary  pro- 
positions as :  /  is  true,  /  is  false,  /  is  certainly  true,  / 
is  experientially  certified,  p  has  been  maintained  by 
Berkeley.     Here  we  are  predicating  various  adjectives 
(the  precise  meaning  of  which  will  be  considered  later) 


SECONDARY  PROPOSITIONS  AND  MODALITY      51 

of  any  given  proposition/;  and  we  define  each  of  these 
propositions — of  which  the  subject-term  is  a  proposition 
and  the  predicate-term  an  appropriate  adjective — as 
secondary.  One  example  may  be  given  which  has  his- 
toric interest.  Take  {A)  the  proposition  'two  straight 
lines  cannot  enclose  a  space' — to  illustrate  a  primary 
proposition;  again  take  {B)  the  proposition  'A  is  es- 
tablished by  experience'  as  a  secondary  proposition; 
and  thirdly  take  (C)  ' B  is  held  by  Mill'  as  a  tertiary 
proposition;  namely — *  It  is  held  by  Mill  that  the 
theorem  that  two  straight  lines  cannot  enclose  a  space 
is  established  solely  by  experience.'  It  is  at  once 
obvious  that,  all  these  three  propositions,  the  primary, 
the  secondary,  the  tertiary,  which  include  the  same 
matter  (viz.  that  expressed  in  the  primary)  might  be 
attacked  or  defended  on  totally  distinct  grounds.  We 
may  defend  the  primary  proposition :  *  two  straight  lines 
cannot  enclose  a  space,'  by  showing  perhaps  that  it  is 
involved  in  the  definition  of  'straight';  again  we  might 
attack  or  defend  the  secondary  proposition :  *  this  geo- 
metrical theorem  is  established  by  experience,'  by 
considering  the  general  nature  of  experience,  and  the 
possibilities  of  proving  generalisations ;  and  lastly,  if  we 
are  to  examine  the  tertiary  proposition,  namely  '  Mill 
held  the  experiential  view  on  the  subject  of  this  geo- 
metrical axiom,'  we  have  only  to  read  Mill's  book  and 
try,  if  possible,  to  understand  what  was  the  precise  view 
that  he  wished  to  maintain. 

§  2.  In  connection  with  a  larger  and  wider  treat- 
ment of  secondary  propositions  in  general,  it  will  be 
useful  here  to  introduce  the  subject  of  Modality.  We 
shall  throughout  speak  of  modal  adjectives,  instead  of 

4—2 


52 


CHAPTER  IV 


modal  propositions',  it  being  understood  that  these 
adjectives  fall  under  the  general  head  of  what  we  have 
called  pre-propositional  adjectives.  We  propose  pro- 
visionally to  include  under  modals  the  adjectives  'true' 
and  'false.'  But  a  question  of  some  interest  arises  as 
to  whether  the  two  very  elementary  cases  ' p  is  true' 
and  'p  is  false'  where/  is  a  proposition  are  legitimate 
illustrations  of  secondary  propositions.  It  may  be  held 
that  the  proposition  'p  is  true'  is  in  general  reducible 
to  the  simple  proposition/;  so  that,  if  this  were  so,  'p 
is  true '  would  only  have  the  semblance  of  a  secondary 
proposition,  and  would  be  equivalent  for  all  ordinary 
purposes  to  the  primary  proposition  /.  It  appears  to 
me  futile  to  enter  into  much  controversy  on  this  point, 
because  it  will  be  universally  agreed  that  anyone  who 
asserts  the  proposition  /  is  implicitly  committing  him- 
self to  the  assertion  that  /  is  true.  And  again  the 
consideration  of  the  proposition  /  is  indistinguishable 
from  the  consideration  of  the  proposition  /  as  being 
true;  or  the  attitude  of  doubt  in  regard  to  the  pro- 
position /  simply  means  the  attitude  of  doubt  as  re- 
gards /  being  true.  These  illustrations,  in  my  view, 
show  that  we  may  say  stricdy  that  the  adjective  true  is 
redundant  as  applied  to  the  proposition  / ;  which  illus- 
trates the  principle,  which  I  have  put  forward,  that  a 
proposition  by  itself  is,  in  a  certain  sense,  incomplete 
and  requires  to  be  supplemented  by  reference  to  the 
assertive  attitude.  Thus  the  assertion  of/  is  equivalent 
to  the  assertion  that  /  is  true  ;  though  of  course  the 
assertump  is  not  the  same  as  the  assertion  that/  is  true. 
The  adjective  true  has  thus  an  obvious  analogy  to  the 
multiplier  d?«^  in  arithmetic:  a  number  is  unaltered  when 


SECONDARY  PROPOSITIONS  AND  MODALITY      53 

multiplied  by  unity,  and  therefore  in  multiplication  the 
factor  one  may  be  dropped ;  and  in  the  same  way  the 
introduction  of  the  adjective  true  may  be  dropped 
without  altering  the  value  or  significance  of  the  pro- 
position taken  as  asserted  or  considered. 

More  interest  attaches  to  the  apparendy  secondary 
proposition  '/  is  false. '  It  certainly  appears  that/-false  is 
indistinguishable  from  not-/,  and  the  majority  of  logicians 
rather  assume  that  not-/  is  on  a  level  with  /,  and  may 
be  at  once  co-ordinated  with  /  as  a  primary  proposition. 
Now  it  appears  to  me  that,  while  /-true  is  practically 
indistinguishable  from  the  primary  proposition  /,  on 
the  other  hand  /-false  is  essentially  a  secondary  pro- 
position, and  can  only  be  co-ordinated  with  primary 
propositions  after  a  certain  change  of  attitude  has  been 
adopted.  This  problem  will  come  up  again  in  the 
general  treatment  of  negation  and  ob version. 

§  3.  We  may  now  turn  to  what  have  been  always 
known  as  modal  adjectives  such  as  necessary,  con- 
tingent, possible,  etc.  The  discussion  of  modality  is 
complicated  rather  unfortunately  owing  to  certain  merely 
formal  confusions  which  have  not  been  explicitly  re- 
cognised. Hence,  before  plunging  into  the  really 
difficult  philosophic  problems,  these  formal  confusions 
must  be  cleared  away.  The  simplest  of  these  occurs  in 
the  controversy  between  those  who  hold  that  contra- 
dictories belong  to  the  same  sphere  of  modality,  and 
those  who  hold  that  they  belong  to  opposite  spheres  of 
modality.  This  controversy  is  resolved  by  explicitly 
realising  the  distinction  between  a  primary  and  a 
secondary  proposition.  Thus  taking,  for  purposes  of 
illustration,  the  antithesis  between  necessary  and  con- 


54 


CHAPTER  IV 


tingent,  we  may  consider  the  primary  proposition  *  It 
is  raining  now'  and  its  contradictory  '  It  is  not  raining 
now';  if  one  of  these  primary  propositions  is  contingent, 
so  also  is  the  other.  But  the  contradictory  of  the 
secondary  proposition  affirming  contingency  of  the 
primary — i.e.  *that  it  is  raining  now  is  contingent' — is 
the  secondary  proposition  which  affirms  necessity  of  the 
primary — i.e. '  that  it  is  raining  now  is  necessary.'  Thus, 
•  in  doubting  or  contradicting  a  secondary  proposition, 
we  use  the  opposite  or  contrary  modal  predicate  ;  but 
in  denying  the  primary  proposition  we  should  attach 
the  same  modal  adjective  to  the  proposition  and  to  its 
contradictory.  There  can  really  be  no  difference  of 
opinion  on  this  subject ;  the  opposition  of  modality  is 
expressed  in  the  secondary  propositions  that  contradict 
one  another ;  the  agreement  in  modality  holds  of  the 
primary  propositions  that  contradict  one  another.  Sum- 
marising :  if  a  given  primary  proposition  is  necessarily 
true,  its  contradictory,  which  is  also  a  primary  pro- 
position, is  necessarily  false;  and  if  a  given  primary 
proposition  is  contingently  true,  its  contradictory,  which 
is  also  a  primary  proposition,  is  contingently  false.  Thus 
in  both  cases  the  contradictory  primary  propositions 
belong  to  the  same  sphere  of  modality.  But  the  con- 
tradictory of  a  secondary  proposition  affirming  necessity 
or  contingency  of  a  primary,  will  be  the  secondary  pro- 
position which  affirms  contingency  or  necessity  of  the 
primary.  Thus  the  contradictories  of  the  secondary  pro- 
positions assert  opposite  modals. 

It  is  necessary  to  enter  into  the  more  philosophical 
aspect  of  modality,  if  only  in  a  preliminary  and  intro- 
ductory way,  because,  apart  from  the  confusion  between 


SECONDARY  PROPOSITIONS  AND  MODALITY      55 

a  secondary  and  a  primary  proposition,  there  is,  it  would 
appear,  considerable  confusion  in  regard  to  the  ter- 
minology adopted  by  different  logicians  or  philosophers 
in  their  treatment  of  modals.  To  do  this  we  feel  bound 
to  reconsider  entirely  the  terminology.  Since  Kant  it 
has  been  customary  to  make  a  three-fold  division,  using 
the  terms  apodictic,  assertoric,  and  problematic;  and  this 
trichotomous  division  at  once  leads  to  some  unfortunate 
confusions.  The  precise  significance  of  assertoric  in 
particular  i^  peculiarly  ambiguous:  thus  the  proposition 
*  2  and  3  make  5 '  as  it  stands,  would  appear  to  be  merely 
assertoric ;  so  that  assertoric  would  include  apodictic  as 
one  of  its  species^  Let  us  then  begin  our  investigation 
without  any  bias  derived  from  the  traditional  terminology. 
§  4.  The  first  antithesis  that  immediately  impresses 
us  in  this  connection  is  that  between  a  certified  and  an 
uncertified  proposition.  A  proposition  which  is  un- 
certified appears  to  be  what  Kant  and  others  have 
sometimes  meant  by  a  problematic  proposition  ;  hence 
we  begin  by  replacing  the  term  '  problematic '  by  the 
term  'uncertified.'  The  contradictory  of  uncertified  is 
certified,  so  that  all  propositions  may  be  divided  into 
the  two  exclusive  classes  of  certified  and  uncertified.  It 
is  of  course  obvious  that  these  terms  are  what  is  called 
relative ;  that  is  to  say,  at  one  stage  in  the  acquisition 
of  knowledge  a  given  proposition  may  be  uncertified, 
while  at  a  later  or  higher  stage,  or  with  increased  oppor- 
tunity of  observation,  etc.,  it  may  become  certified.   The 

^  This  confusion  is,  of  course,  due  merely  to  the  failure  to  dis- 
tinguish between  a  primary  proposition  as  such  and  a  secondary.  It 
is  totally  independent  of  any  question  as  to  what  the  adjectives 
assertoric  and  apodictic  mean  respectively. 


56 


CHAPTER  IV 


"S 


distinction  therefore  is  of  course  not  permanent  or 
absolute,  but  temporal  and  relative  to  individuals  and 
their  means  of  acquiring  knowledge.  It  might  be  held 
that  such  distinctions  should  be  excluded  from  Logic ; 
but  this,  in  our  opinion,  is  unsound,  in  as  much  as 
reference  to  the  mental  powers  and  the  individual 
opportunities  of  acquiring  knowledge  turns  out  in  many 
discussions  to  be  a  most  essential  topic  for  logical  treat- 
ment. The  whole  doctrine  of  probability  hinges  upon 
our  realising  the  changeable  or  relative  opportunities 
and  means,  which  differ,  from  one  situation  to  another, 
in  the  extent  of  attainable  knowledge.  The  further  dis- 
cussion then  of  uncertified  propositions  will  later  intro- 
duce the  logical  topic  of  probability.  Returning  to 
certified  propositions,  a  distinction  is  required  according 
as  the  given  proposition  is  certified  as  true  or  certified 
as  false ;  and  thus  we  have  a  triple  division :  uncertified, 
certified  as  true,  and  certified  as  false.  But  for  most 
purposes  this  latter  distinction  is  unnecessary,  because 
for  the  given  proposition  that  has  been  certified  as  false 
we  might  substitute  the  contradictory  proposition  that 
has  been  certified  as  true.  It  would  be  enough  there- 
fore to  use  the  two  divisions  uncertified  and  certified, 
understanding  by  certified  *  certified  as  true.' 

§  5.  The  above  division  leads  to  a  fundamentally 
important  subdivision  under  the  term  'certified';  for  we 
must  recognise,  in  epistemology  or  general  philosophy, 
that  there  are  essentially  different  principles  or  modes 
by  which  the  truth  of  a  proposition  may  be  certified  ; 
and  a  rough  two-fold  classification  will  conveniently 
introduce  this  subject:  thus  we  may  contrast  a  proposi- 
tion whose  truth  is  certified  by  pure  thought  or  reason 


SECONDARY  PROPOSITIONS  AND  MODALITY      57 

with  a  proposition  which  is  certified  on  the  ground  of 
actual  experience.  Briefly  we  shall  call  these  two  classes 
'  formally  certified  *  and  *  experientially  certified.'  The 
range  of  these  two  modes  of  certification  will  be  a  matter 
of  dispute :  some  philosophers  hold  that  all  the  principles 
and  formulae  of  logic,  and  all  those  of  arithmetic  and 
mathematics,  are  to  be  regarded  as  certified  by  pure 
thought  or  reason.  This  gives  perhaps  the  widest  range 
for  the  propositions  that  may  be  said  to  be  formally 
certified.  But  even  amongst  these,  we  may  have  to 
distinguish  those  which  Aave  been  formally  certified, 
from  amongst  the  entire  range  which  may  be  regarded 
as  formally  certifiable.  Others  would  hold  that  many 
mathematical  principles,  such  as  those  of  geometry,  can 
only  be  certified  by  an  appeal  to  sense-perception — a 
form  of  experience;  and  thus  the  limits  to  be  ascribed  to 
the  range  of  formal  certification  would  open  up  serious 
controversy.  Again,  on  the  other  hand,  the  range  of 
propositions  immediately  certifiable  by  experience  raises 
serious  problems.  Some  may  hold  that  the  only  truths 
guaranteed  by  mere  experience  are  the  characterisations 
of  actual  sense-impressions  experienced  by  the  thinker 
at  the  moment  in  which  he  asserts  the  proposition ;  many 
would  extend  this  to  judgments  on  the  individual's  past 
experiences  revived  in  memory;  but  the  most  universally 
understood  range  of  experientially  certified  propositions 
is  still  wider:  it  would  include  sense-perceptions,  and 
observations  of  physical  phenomena,  and  even  judg- 
ments on  mental  phenomena, — these  supplying  the 
required  data  for  science  in  general.  We  will  not  then 
profess  to  draw  the  line  precisely  between  propositions 
that  are  to  be  regarded  as  formally  certifiable  and  those 


58 


CHAPTER  IV 


that  are  to  be  regarded  as  experientially  certifiable  ; 
but  there  is  one  explanation  of  the  relation  between 
these  two  classes  which  will  probably  be  admitted  by 
alf;  namely,  that  propositions  which  are  admittedly 
based  on  experience,  will  also  involve  processes  of 
thought  or  reasoning,  and  that  therefore  no  propositions 
of  any  importance  are  based  upon  experience  alone  ; 
since  an  element  of  thought  or  reason  enters  into  the 
certification  of  all  such  propositions.  This  leads  to 
a  simple,  more  precise  definition  of  the  antithesis — 
formal  and  experiential :  while  we  define  a  formally 
certifiable  proposition  as  one  which  can  be  certified  by 
thought  or  reason  alone,  we  do  not  define  experiential  pro- 
positions as  those  which  can  be  certified  by  experience 
alone,  but  rather  as  those  which  can  only  be  certified 
with  the  aid  of  experience.  In  this  way  we  imply  that 
experience  alone  would  be  inadequate \ 

A  certain  relation  between  the  two  antithetical 
modals,  formal  and  experiential,  will  be  found  to  apply 
over  and  over  again  to  other  antitheses  in  the  charac- 
teristics of  propositions.  It  may  be  illustrated  by 
reference  to  the  syllogism.  Thus  a  certain  syllogism 
may  contain  one  formal  premiss  and  one  experiential 
premiss  ;  and  the  conclusion  deducible  from  these  two 
premisses  must  be  called  experiential,  because  it  has  been 
certified  by  at  least  one  experiential  premiss.  To  put  it 
otherwise,  if  all  the  premisses  of  an  inference  were  formal, 

^  Even  this  distinction  requires  amendment;  for  it  may  be  main- 
tained that  just  as  experience  alone  can  certify  nothing,  so  thought 
alone  can  certify  nothing.  Thus  formal  certification  would  coincide 
with  what  requires  only  experience  in  general  (to  use  Kantian  termin- 
ology) whilst  experiential  certification  would  involve  in  addition  special 
or  particular  experience. 


SECONDARY  PROPOSITIONS  AND  MODALITY      59 

the  conclusion  would  be  formal ;  but  if  only  one  premiss 
is  experiential  (even  though  the  others  may  be  formal), 
the  conclusion  must  be  experiential.  This  particular 
characteristic  of  the  syllogism  is  not  arbitrary,  but 
follows  from  the  common  understanding  of  what  is 
meant  by  *  experientially  certified,'  namely  something 
which  could  not  be  certified  without  experience, — not 
something  which  could  be  certified  by  experience  alone. 
§  6.  One  of  the  chief  sources  of  confusion  is  the 
use  of  the  term  *  necessary'  in  various  different  senses 
as  an  adjective  predicable  of  propositions.  It  has  some- 
times been  said  that  all  propositions  should  be  con- 
ceived as  necessary;  in  the  sense  that  the  asserter  of  a 
proposition  represents  to  himself  an  objective  ground 
or  reference  to  which  he  submits  and  which  restrains 
the  free  exercise  of  his  will  in  the  act  of  judgment. 
This  contention  is  indisputable,  and  may  be  regarded 
as  one  of  the  many  ways  in  which  the  nature  of  judg- 
ment or  assertion  as  such  may  be  philosophically  ex- 
pounded. But  obviously  necessity  as  so  conceived  can- 
not serve  as  a  predicate  for  distinguishing  between 
propositions  of  different  kinds.  We  pass,  therefore,  to 
the  next  and  more  usual  meaning  of  the  term  necessary  ^ 
which  will  perhaps  best  be  indicated  by  a  quotation  from 
Kant:  'Mathematical  propositions  are  always  judg- 
ments a  priori  and  not  empirical,  because  they  carry 
with  them  the  conception  of  necessity,  which  cannot  be 
given  by  experience.'  Here  necessary  is  opposed  to 
empjrical ;  and  the  antithesis  that  Kant  has  in  view 
coincides  approximately  with  that  between  xh^  formally 
certified  and  the  experientially  certified  (as  I  have  pre- 
ferred to  express  it).     But  still  another  meaning  has 


ft. 


y< 


6o 


CHAPTER  IV 


been  attached  to  the  term  necessary,  viz.,  that  according 
to  which  the  necessary  is  opposed  to  the  contingent. 
If,  however,  the  term  contingent  is  interpreted  as  equiva- 
lent to  (what  I  have  called)  experientially  certified,  then 
we  might  agree  that  necessary  should  be  interpreted  as 
equivalent  to  formally  certified;  and  in  this  case  we 
should  not  have  found  a  third  meaning  to  the  term. 
The  question  therefore  arises  whether  a  use  can  be  found 
j  for  the  antithesis  'necessary'  and  'contingent,'  within 
I  the  sphere  of  the  experientially  certified.   Now  it  has  been 
maintained    as   a  fundamental  philosophical  postulate 
that  'All  that  happens  is  necessitated ' ;  and  this  may  be 
taken  as  equivalent  to  saying  that  'Nothing  that  hap- 
pens is  contingent."     It  should  here  be  pointed  out  that 
this  contention  is  to  be  clearly  distinguished  from  the 
view  that  'All  judgments  or  propositions  are  necessary! 
For  the  necessity  ascribed  to  judgments  is  conceived  as 
a  compulsion  exercised  by  the  objective  or  real  upon 
the  thinker;    whereas  the  necessitation  attributed    to 
events  is  conceived  (more  or  less  metaphorically)  as  a 
compulsion  exercised  by  nature  as  a  unity  upon  natural 
phenomena  as  a  plurality.     The  former  necessity  is  so 
to  speak  objectivo-subjective;  the  latter  objectivo-ob- 
jective.    But  an  elementary  criticism  must  be  directed 
against  the  use  made  of  the  postulate  'All  that  happens 
is  necessitated'  to  deduce  that  there  is  no  proper  scope 
for  the  term  contingent.    For  we  inevitably  conceive  of 
that  which  happens  as  being  necessitated  by  something 
else  that  happens  in  accordance  with  (what  is  popularly 
called)  a  law  of  nature.    In  other  words,  the  laws  of 
nature  taken  alone  do  not  necessitate  any  event  what- 
ever;  we   should    have  rather  to  say  that  a  law  of 


SECONDARY  PROPOSITIONS  AND  MODALITY      6i 

nature  necessitates  that  the  happening  of  some  one  thing 
should  necessitate  the  happening  of  a  certain  other 
thing.  Hence,  I  should  propose  that  nomic  (from  pofioq, 
a  law)  should  be  substituted  for  necessary  as  contrasted 
with  contingent.  Thus  a  nomic  proposition  is  one  that 
expresses  a  pure  law  of  nature  ;  and  a  contingent  propo- 
sition is  one  that  expresses  a  concrete  event.  In  this  way 
we  have  eliminated  the  ambiguous  term  necessary,  and 
have  substituted  y^r;;^a//v  certified  when  the  term  is  op- 
posed to  experientially  certified;  and  ;e(?;;^^V  when  opposed 
to  contingent.  Finally  the  \.exvci possible  must  be  coupled 
with  the  word  necessary  in  its  three  usages.  For  'pos- 
sible' has  three  obviously  distinct  meanings:  (i)  what 
is  not  known  to  be  false;  i.e.  what  does  not  contradict 
the  necessary  in  the  first  sense,  applicable  to  all  asser- 
tions ;  (2)  what  does  not  conflict  with  any  formally  certi- 
fied proposition,  i.e.  with  any  proposition  necessary  in 
the  second  sense;  (3)  what  does  not  conflict  with  any 
law  of  nature,  i.e.  with  any  proposition  necessary  in  the 
third  sense.  The  word  'possible'  in  these  three  senses 
may  be  distine^uished  respectively  as  '  the  epistemically 
possible,'  *the  formally  possible'  and  'the  nomically  I 
possible.' 

§  7.  It  will  now  be  apparent  that  the  antithesis 
between  nomic  and  contingent  is  of  a  totally  different 
nature  from  that  between  certified  and  uncertified,  or 
between  the  different  modes  of  certification.  The  latter 
has  been  called  subjective,  the  former  objective;  but 
the  terms  epistemic  and  constitutive  are  preferable :  for 
the  characteristics  'nomic'  and  'contingent'  apply  within 
the  content  of  the  proposition,  and  are  therefore  properly 
to  be  regarded  as  constitutive ;  whereas  the  character- 


62 


CPL\PTER  IV 


istics  *  certified'  and  'uncertified'  apply  to  the  relation 
of  the  proposition  to  the  thinker,  and  should  therefore 
be  called  epistemic.  Taking  for  example,  the  propo- 
sition *  Nature  is  uniform ' :  if  this  is  held  to  be  neces- 
sary in  the  sense  that  our  reason  alone  establishes 
its  truth,  then  the  attribution  of  necessity  is  in  this 
case  of  the  same  kind  as  what  we  have  called  formally 
certified  and  is  thus  epistemic.  But  the  necessity  in- 
volved in  the  laws  of  nature  is  generally  attributed  to 
Nature  itself,  and  not  merely  to  our  grounds  for  assert- 
ing such  uniformity:  and  is  thus  constitutive.  Thus,  if 
we  say,  as  a  specific  example  of  the  necessity  attributed 
to  Natures  processes,  that  'bodies  attract  one  another 
in  obedience  to  the  necessities  of  nature,'  this  statement 
is  quite  independent  of  any  view  we  may  hold  as  to 
the  reasonable  grounds  for  asserting  the  fact  of  universal 
gravitation.  In  short,  referring  back  to  the  distinction 
between  the  fact  and  the  proposition,  such  modals  as 
certified  and  uncertified  are  adjectives  directly  character- 
ising the  proposition,  whereas  modals  of  the  other  kind, 
typified  by  nomic  and  contingent,  directly  characterise 
the/acL 

§  8.  It  remains  now  to  introduce  a  certain  familiar 
distinction  amongst  propositions  not  included  in  the 
understood  meaning  of  modal,  viz.  that  between  real 
and  verbal  These  terms  were  used  by  Mill,  and  are 
generally  understood  as  equivalent  to  Kant's  terms 
'synthetic'  and  'analytic'  Mill's  point  of  view  is  very  dif- 
ferent from  Kant's,  for  Mill  is  thinking  of  the  nature  of 
language,  of  the  definition  of  words,  etc.,  while  Kant  is 
thinking  of  ideas  and  the  various  constructive  acts  of 
thought.    Mill's  usage  is  more  easy  to  expound  than 


SECONDARY  PROPOSITIONS  AND  MODALITY      63 

Kant's,  and  gives  rise  to  less  serious  conflict  of  view. 
A  verbal  proposition  is  one  which  can  be  affirmed  from 
a  mere  knowledge  of  the  meanings  of  words  and  their 
modes  of  combination ;  a  real  proposition,  on  the  other 
hand,  requires  for  its  acceptance,  not  only  a  knowledge 
of  the  meanings  of  words,  but  also  a  knowledge  of  mat- 
ters of  fact.  We  may  therefore  note  the  same  relation 
between  verbal  and  real  as  between  formal  and  experi- 
ential': namely,  that  two  premisses,  both  of  which  are 
verbal,  can  only  yield  a  verbal  conclusion;  and  that 
a  single  real  premiss,  even  though  joined  with  any 
number  of  verbal  premisses,  will  impose  upon  the  con- 
clusion its  own  character  as  real. 

The  ^definition  so  far  given  of  verbal  propositions 
seems  fairly  clear;  it  is  therefore  surprising  that  it 
should  have  proved  a  stumbling-block  to  some  logi- 
cians. The  people  who  have  raised  difficulty  on  this 
point  are  those  who  have  preferred  the  Kantian  terms 
'analytic'  and  'synthetic'  in  place  of  Mill's  terms  'ver- 
bal' and  'real':  ('analytic'  Kant  illustrates  by  the  pro- 
position 'Material  bodies  are  extended,'  'synthetic'  by 
the  proposition  'Material  bodies  attract  one  another'). 
The  controversy  has  arisen  through  a  tacit  confusion 
between  'verbal  or  analytic'  and  'familiar'  on  the  one 
hand,  and  between  'real  or  synthetic'  and  'unfamiliar' 
on  the  other  hand,  due  to  the  kind  of  examples  chosen 
to  illustrate  each  type  of  proposition.  This  confusion  y^ 
is  apparent  in  the  well-known  dictum  of  Bradley — 'that 
synthetic  judgments  are  analytic  in  the  making' — where 
it  is  clear  that  by  a  'synthetic  judgment'  he  means  the 
newly-constructed  proposition,  and  by  'in  the  making,' 

^  See  above,  last  paragraph  of  §  5. 


64 


CHAPTER  IV 


the  process  of  rendering  the  proposition  familiar.  But, 
it  needs  only  a  little  reflection  to  show  that  familiarity 
wijth  a  matter  of  fact  does  not  render  the  proposition 
which  expresses  such  fact  verbal  or  analytic ;  nor  does 
unfamiliarity  with  the  meanings  of  words  render  a  pro- 
position which  explains  such  meaning  real  or  synthetic: 
a  proposition  about  the  meanings  of  words  is  verbal, 
and  a  proposition  about  matters  of  fact  is  real,  whether 
the  hearer  is  unfamiliar  or  familiar  with  the  words  or 
with  the  facts\  Thus  the  proposition  '7  and  5  make  12* 
is  familiar  enough,  but  whether  or  not  it  is  verbal  (or 
analytic)  has  absolutely  nothing  to  do  with  its  familiarity; 
on  the  other  hand,  a  technical  definition  given  by  a 
scientist  will  probably  be  quite  unfamiliar,  but  if  the 
scientist  puts  it  forward  as  an  expression  of  his  inten- 
tion  to  use  a  word  with  a  certain  significance,  the  pro- 
position which  states  his  intention  is  verbal,  although  it 
is  ipso  facto  unfamiliar. 

Perhaps  a  better  way  of  indicating  the  nature  of  a 
verbal  proposition,  is  to  say  that  it  is  not  quite  what  is 
ordinarily  meant  by  a  proposition;  that  is,  as  verbal,  it 
cannot  strictly  be  said  to  be  either  true  or  false,  because 
it  does  not  declare  a  fact,  but  rather  expresses  an 
intention,  a  command,  or  a  request.  The  technical 
scientist  puts  forward  his  definitions  in  this  spirit,  when 
he  asks  readers  to  allow  him  to  use  a  term  with  a  cer- 

1  An  important  explanation  of  all  this  should  be  given.  What 
Bradley  means  by  *'an  analytic  judgment"-not  "a  verbal  proposi- 
tion"—is  a  judgment  that  could  be  discovered  by  mtrospective 
analysis,  so  that  his  pronouncement  is  an  obvious  truism.  But  it  is 
strange  that  he  does  not  perceive  that  this  is  not  in  the  least  the  same 
as  what  Kant  meant.  Kant's  distinction  is  epistemological,  Bradley's 
merely  psychological. 


SECONDARY  PROPOSITIONS  AND  MODALITY      65 

tain  signification  which  is  explained  by  his  definition. 
Thus  a  verbal  proposition  is  neither  true  nor  false,  be- 
cause it  is  properly  expressed,  not  in  the  indicative,  but 
in  the  imperative  or  other  similar  mood.  But  if  by 
a  verbal  proposition  is  meant  one  that  assigns  the 
meaning  of  a  word  as  conventionally  used  in  any  wider 
or  narrower  context,  then,  inasmuch  as  the  proposition 
asserts  the  fact  that  such  or  such  is  the  convention,  it 
must  be  either  true  or  false. 

§  9.  At  the  beginning  of  this  chapter  we  defined  a 
secondary  proposition  as  one  that  predicates  one  or 
other  of  the  adjectives  significantly  predicated  of  a 
proposition  as  such.  We  proceeded  to  consider  in  turn 
different  kinds  of  adjectives  that  are  thus  predicable  : 
this  has  led  to  a  discussion  of  modal  adjectives,  and  has 
included  in  particular  a  consideration  of  the  adjectives 
true  and  false,  and  finally  of  the  predicates  *  verbal'  or 
'analytic'  and  'realtor  *  synthetic' 


J.  L. 


66 


NEGATION 


67 


CHAPTER  V 


NEGATION 


§  I.  Under  the  general  problem  of  the  nature  of 
negation  we  may  begin  by  considering  the  particular 
form  of  negation  which  has  been  called  '  pure  negation.' 
There  appear  to  be  several  different  meanings  attached 
to  the  notion  oipure  negation :  it  may  mean  the  simple 
attitude  of  rejection,  as  opposed  to  that  of  acceptance, 
towards  a  proposition  taken  as  a  unit  and  without  further 
analysis.  Such  negation  may  be  called  pure,  because 
the  negative  element  does  not  enter  within  the  content  of 
the  assertum,  but  expresses  merely  a  certain  mental  atti- 
1  tude  to  the  proposition  itself.  According  to  this  definition 
of  pure  negation,  the  judgment  which  may  be  called 
purely  negative  has  as  its  object  precisely  what  I  have 
called  a  secondary  proposition  in  my  previous  discussion 
as  to  whether  the  statement ' p  is  false '  is  to  be  regarded 
as  primary  or  as  secondary.  When  then  we  enquire  as 
to  the  importance  or  the  relevance  of  pure  negation, 
we  may  be  raising  the  question  whether  a  judgment 
expressed  in  this  purely  negative  form  really  ever  re- 
presents a  genuine  attitude  of  thought.  No  doubt  there 
are  not  many  cases  in  which  this  negative  attitude 
towards  an  assertum  taken  as  a  unit  could  be  illustrated; 
but  we  may  at  least  insist  that,  when  some  assertum  is 
proposed  which  can  be  clearly  conceived  in  thought, 
and  yet  repels  any  attempt  to  accept  it,  then  the  attitude 


towards  such  an  assertum  to  which  our  thinking  process 
has  led  us  is  strictly  to  be  called  that  of  pure  negation. 
For  example,  the  proposition  *  Matter  exists '  may  appear 
to  some  philosophers  to  have  in  it  a  sufficiently  clear 
content  to  enable  them  to  reject  it,  without  their  having 
in  mind  any  correspondingly  clear  substitute  which  they 
can  accept.  In  this  case  their  mental  attitude  towards 
the  proposed  assertum  may  be  properly  called  one  of 
mere  negation ;  since  the  only  positive  element  involved 
is  the  conceived  content  of  the  proposition  rejected. 

But  the  term  pure  negation  is  more  generally  applied 
where  a  predicate  is  denied  of  some  subject  within  the 
proposition.  Under  this  head,  the  case  where  negation 
would  seem  to  be  quite  pure  may  be  illustrated  by  a 
proposition  like — 'Wisdom  is  not  blue.'  Such  a  pro- 
position would  have  purpose  only  in  a  logical  context 
where  we  are  pointing  out  that  certain  types  of  adjec- 
tive cannot  be  predicated  of  certain  types  of  substantive. 
A  more  common  case  which  leads  to  a  purely  negative 
form  of  predication,  is  where,  for  instance,  a  distant 
object  of  perception,  is  considered  as  to  whether  it  is 
blue  or  of  some  other  colour,  or  as  to  whether  it  is  a 
man  or  some  other  material  body.  Towards  this  pro- 
posed assertum — that  it  is  blue,  or  that  it  is  a  man — our 
attitude  may  be  that  of  mere  denial,  in  the  sense  that 
we  are  perfectly  clear  what  it  is  not,  but  we  are  not 
correspondingly  clear  as  to  what  it  is.  We  may  admit 
that  a  judgment  which  in  this  sense  is  merely  negative 
and  without  any  positive  content  is  rare,  since  when  we 
deny  of  a  flower  that  it  is  red,  we  are  at  least  judging 
that  it  has  some  colour,  and  similarly  when  we  deny  of 
something  in  sight  that  it  has  the  shape  of  a  man  we  are 

5—2 


68 


CHAPTER  V 


NEGATION 


69 


7' 


at  least  judging  that  it  has  some  shape,  and  this  consti- 
tutes a  positive  element  in  our  judgment.  The  above 
examples  illustrate  two  applications  of  the  notion  of 

r  negation:  first,  in  denying  the  proposition  as  a  whole, 
and  again  in  denying  that  an  adjective  of  a  certain  type 
can  be  predicated  of  a  certain  type  of  substantive,  where 

2-  the  positive  element  is  evanescent;  and  secondly,  in 
denying  the  more  specific  predicate  proposed  for  a 
substantive  while  tacitly  asserting  some  wider  predicate 
under  which  it  falls,  where  a  positive  element  is  properly 
to  be  recognised. 

Some  logicians,  going  one  step  further,  have  asserted 
that,  in  denying  an  object  to  be  red,  not  only  is  the 
generic  adjective  colour  a  positive  factor  in  the  judg- 
ment, but  that  some  specific  colour  other  than  red  is 
tacitly  affirmed :  that  is,  they  hold  that  we  cannot  deny 
unless  we  have  some  positive  determinate  ground  for 
our  denial.  But  this  reason  for  asserting  the  universal 
presence  of  a  positive  factor  in  judgment  must  not  be 
confused  with  the  former;  for  it  is  one  thing  to  say 
that  the  denying  of  any  proposed  adjective  involves 
the  affirming  of  some  other  adjective  of  the  same 
generic  kind,  and  another  thing  to  say  that  it  involves 
the  affirming  of  a  specific  adjective.  While  admitting 
the  first,  I  reject  the  view  that  in  denying  red  we  are 
affirming  say  green  or  blue  as  the  case  may  be,  on  the 
ground  that  it  involves  a  confusion  between  what  is 
necessarily  determined  in  fact  with  what  may  or  may 
not  be  determinate  in  our  knowledge  of  fact.  There 
are  countless  cases  of  our  denying  a  certain  proposed 
adjective  in  which,  while  we  know  that  some  determinate 
adjective  can  be  truly  applied,  yet  we  do  not  know 


ivhich  determinate  adjective  is  to  be  substituted  for  that 
rejected.  The  most  obvious  illustration  is  in  predications 
of  place  and  time:  thus  we  may  say  '  Mr  Smith  is  not 
now  in  this  room,'  and,  knowing  that  Mr  Smith  is  alive, 
we  know  that  in  the  necessities  of  nature  he  must  be  in 
some  other  determinate  place.  Thus  we  may  in  a  rapid 
survey  discover  the  absence  of  any  object  within  a  given 
place,  independently  of  any  knowledge — by  observation 
or  otherwise — of  its  presence  in  some  other  place ;  and 
this  is  sufficient  to  dispose  of  the  contention  that  there 
must  be  positive  ground  for  a  negative  judgment.  In 
fact  the  strictly  negative  form  of  judgment  is  relevant 
for  purposes  of  further  development  of  thought,  whether 
we  are  able  to  assert  an  opposed  positive,  or  know  only 
that  som^e  opposed  positive  could  be  affirmed  if  our 
knowledge  were  further  extended.  What  is  obviously 
true  of  time  or  place  predications  is  also,  though  not 
always  so  obviously,  true  of  qualitative  predicates  such 
as  colour  or  tone  :  for  instance,  we  may  deny  that  a 
certain  sound  is  that  of  a  piano,  because  of  our  familiarity 
with  that  instrument,  without  being  able  to  define  the 
kind  of  musical  instrument  from  which  the  sound  pro- 
ceeds, owing  perhaps  to  our  unfamiliarity  with  other 
instruments;  although  we  may  know,  first,  that  it  is  a 
musical  sound,  and  secondly  on  quite  general  grounds 
that  it  must  come — not  from  any  instrument  whatever — 
but  from  some  determinate  kind  of  instrument. 

§  2.  Having  distinguished  some  of  the  different  ways 
in  which  the  phrase  pure  negation  may  be  understood, 
we  will  briefly  examine  the  dictum  that  pure  negation 
has  no  significance.  It  may  perhaps  be  at  once  said  that 
this  dictum  is  itself  purely  negative,  and  that  therefore 


70 


CHAPTER  V 


NEGATION 


71 


anyone  who  maintains  its  significance  has  committed 
himself  to  a  contradiction.  A  more  serious  treatment 
of  the  contention  shows  that  for  the  word  '  significance  ' 
we  should  substitute  'having  value'  or  'importance' 
or  'relevance  to  a  specific  purpose.'  The  purpose,  for 
instance,  of  the  above  negatively  expressed  dictum  is 
to  oppose  some  other  philosophers  who  have  attributed 
a  false  value  or  importance  to  the  negative  judgment. 
It  will  be  seen  that  the  whole  question  hinges  on  the 
meaning  to  be  attached  to  the  word  'significance.'  A 
form  of  words  maybe  said  to  he  a6so/u^e/ynon-sigmfiC3Lnt 
when  they  fail  to  convey  any  precise  content  for  thought- 
construction.  This  failure  of  a  phrase  to  convey  meaning 
may  be  due  either  to  the  substantial  components  them- 
selves or  to  their  mode  of  combination ;  thusitis  a  merely 
verbal  expression  that  may  be  said  to  have  or  not  have 
significance  for  thought  in  this  absolute  sense.  But  in 
attributing  non-significance  to  a  judgment  apart  from 
its  verbal  expression,  the  most  probable  meaning  in- 
tended is  that  it  does  not  represent  any  actual  process 
in  thought.  But  any  of  the  examples  taken  above  go 
to  show  that  the  purely  negative  judgment  cannot  be 
universally  charged  with  non-significance  in  this  sense. 
§  3.  We  have  considered  in  turn,  first  the  proposition 
as  a  whole  unanalysed ;  secondly,  the  predication  of  an 
adjective  of  a  given  subject-term;  and  we  now  turn  to 
the  subject-term  itself,  apart  from  the  adjective  predi- 
cated, and  raise  the  question  whether  any  proposition 
can  have  significance  in  case  there  is  no  real  thing 
corresponding  to  the  subject-term,  although  theremaybe 
a  word  or  phrase  used  professedly  to  denote  such  thing. 
Now  I  have  regarded  the  substantive,  which  is  ultimately 


the  subject  in  all  propositions,  as  a  determinandum — 
that  is,  as  something  given  to  be  determined  in  thought ;  if 
then  there  is  nothing  given  to  be  so  determined  corre- 
sponding to  the  word  or  phrase  by  which  we  intend  a 
certain  substantive,  then  what  becomes  of  the  propo- 
sition .'*  Consider  for  example  the  propositions :  '  An 
integer  between  3  and  4  is  prime,'  and  again  '  An  integer 
between  3  and  4  is  composite.'  It  must  be  said  that 
neither  of  these  propositions  is  true.  Now  since  every 
integer  is  either  prime  or  composite,  it  can  be  at  once 
seen  that  any  proposition  predicating  an  adjective  of  the 
subject '  an  integer  between  3  and  4 '  must  be  false,  even 
though  the  adjective  is  appropriate  to  integers  as  such. 
This  statement  needs  only  the  qualification  that  we  may 
correctly  predicate  of  an  integer  between  3  and  4  that 
it  is  greater  than  3  and  less  than  4 ;  this,  however,  is 
not  a  genuine  proposition  but  one  that  is  implied  in  the 
meaning  of  the  subject-term,  and  is  thus  merely  verbal. 
We  conclude  then  that  of  such  a  subject-term  as  '  an 
integer  between  3  and  4  '  no  adjective  can  be  truly  pre- 
dicated in  a  real  or  genuine  proposition. 

We  may  therefore  contrast  two  cases  of  a  subject- 
term  S\  (i)  where  5  is  such  that  some  adjective  can 
be  truly  predicated  of  it  in  a  genuine  proposition,  and 
(2)  where  6*  is  such  that  no  adjective  can  be  truly  pre- 
dicated of  it  in  a  genuine  proposition.  These  two  cases 
may  be  briefly  expressed — '  5  is  '  and  '  S  is  not.'  The 
significance  of  these  two  propositions  is  brought  out  in 
considering  the  process  technically  known  as  obversion. 
The  fundamental  problem  of  obversion  I  will  symbolise 
as  the  problem  of  passing  from  '  5  is-not  /* '  to  '  5  is 
non-/^.'    Here,  when  we  hyphen  the  negative  with  the 


C/y6->''*rv*'t«-W 


72 


CHAPTER  V 


NEGATION 


73 


copula,  I  understand  it  to  mean  that  the  proposition 
*  5'  is  /» '  as  a  unit,  is  asserted  to  be  false.  But  when 
we  hyphen  the  negative  with  the  predicate,  we  are 
^affirming  of  the  subject  5  the  kind  of  predicate  called 
negative;  in  other  words  '  6"  is  non-P'  is  an  affirmative 
proposition  containing  a  negative  predicate,  while  '  5  is- 
not  P '  is  a  negative  proposition  in  the  sense  that  the 
'■  attitude  of  negation  applies  to  the  proposition  as  a  whole. 
Now  this  transformation  from  the  negative  proposition 
to  the  positive  assertion  of  a  negative  predicate,  has 
been  assumed  as  almost  trifling,  and  as  only  too  obvious; 
but  I  would  wish  at  once  to  raise  the  question  as  to  the 
condition  necessary  for  the  validity  of  this  process, 
called  obversion,  in  its  fundamental  form. 

As  we  have  already  stated,  the  incomplete  pro- 
position '  5  is '  really  means,  *  5*  denotes  something 
of  which  some  adjective  may  be  predicated  truly  in  a 
proposition  not  merely  verbal.'  Thus  the  scheme  by 
which  I  express  the  condition  under  which  obversion 
is  valid,  is  to  add  to  the  explicit  negative  premiss  '  S 
is-not  P,'  the  additional  premiss  '  S  is,'  from  which  we 
may  validly  infer  the  affirmative  conclusion  '  S  is  non- 
PJ  The  incomplete  form  of  proposition  '  5  is '  means 
that  5*  has  some  character  which  may  be  predicated  of 
it,  without  defining  what  character  can  be  positively 
asserted.  The  conclusion  *  6'  is  non-P'  means  that  we 
predicate  of  5  a  character,  determined  so  far  as  that  it  is 
an  opponent  of  the  proposed  character  P,  but  otherwise 
indeterminate.  An  illustration  from  history  will  show 
how  this  process  may  be  applied.  Thus  the  name 
William  Tell  is  the  name  of  a  historical  character  about 
whose  existence  there  appears  to  be  doubt.    In  denying 


any  proposition  which  predicates  an  adjective  such  as 
'submissive'  of  the  subject  William  Tell,  we  could  not 
validly  predicate  of  him  the  contrary  adjective  '  defiant,' 
unless  we  were  able  first  to  assert  that  Tell  is,  in  the 
sense  we  have  explained. 

The  problem  of  the  obversion  of  a  singular  propo- 
sition is  the  same  as  that  of  formulating  accurately  the 
contradictory  of  a  singular  proposition.  Thus,  in  showing 
that,  in  order  to  pass  from  the  denial  (or  contradictory) 
of  '  kS  is  /*'  to  the  affirmation  *  5  is  non-P*  we  require 
the  additional  datum  '  5  is,'  we  have  indicated  that 
neither  of  the  propositions  *  5  is  P'  and  '  S  is  non-P ' 
would  be  true,  in  the  case  that  *  5*  is '  were  not  true. 
In  other  words,  the  two  propositions  ' S\^  P'  and  ' S 
is  non-/^ '  are  not  properly  contradictories.  The  contra- 
dictory of  *  kS  is  /* '  should  be  formulated  in  the  alterna- 
tive proposition  *  Either  S  is-not  or  S  is  non-P ' ;  as  also 
the  contradictory  of  *  6*  is  non-P '  in  the  alternative 
proposition  '  Either  S  is-not  or  S  is  P!  Thus,  in  our 
historical  illustration,  neither  of  the  two  propositions  *  A 
certain  man  named  William  Tell  submitted  to  the 
Austrians '  and  '  A  certain  man  named  William  Tell 
defied  the  Austrians '  would  be  true,  if  it  were  the  case 
that  there  was  no  such  person  as  William  Tell;  and 
hence  the  proper  contradictories  of  the  two  propositions 
must  be  respectively  expressed  in  the  alternative  forms : 
'  Either  there  was  no  such  person  as  Tell  or  he  (Tell) 
defied  the  Austrians,'  and  *  Either  there  was  no  such 
person  as  Tell  or  he  (Tell)  submitted  to  the  Austrians.' 

§  4.  To  illustrate  the  significance  of  this  view  we 
must  consider  the  different  types  of  cases  in  which  a 
proposition  of  the  form — *  5  is ' — can  be  truly  asserted. 


74 


CHAPTER  V 


NEGATION 


75 


In  every  case,  the  term  5  must  have  sufficiently  de- 
terminate meaning,  to  give  rise  to  the  alternative  pro- 
positions '  5  is '  or  '  5*  is  not ' ;  the  question  could  not 
arise  if  S  were  treated  as  a  mere  symbol  without  sig- 
nificance. When  this  is  agreed,  it  will  be  found  that 
any  apparent  variations  in  the  meaning  of  the  word  '  is,' 
will  in  reality  be  variations  in  the  kinds  of  substantive 
category  to  which  the  name  .S  is  understood  to  apply. 
For  instance,  let  us  take  the  names  of  substantives 
under  the  category  of  number.  We  may  say  on  the 
positive  side  that  the  number  3  is.  This  will  mean 
that  some  true  adjectives  can  be  predicated  of  the 
number  3,  beyond  those  which  might  be  held  as  merely 
involved  in  the  definition  or  connotation  of  the  word  3 ; 
thus,  if  we  should  define  3  as  meaning  2  +  1,  the  state- 
ment that  the  number  3  has  the  characteristic  expressed 
by  2  +  I  would  be  purely  verbal.  But  the  number  3, 
we  say,  is  such  that  an  indefinite  number  of  other  ad- 
jectives, not  included  in  its  definition,  can  be  truly  pre- 
dicated, as  for  instance  that  3  is  prime  or  that  3  is  a 
factor  of  12.  Contrast  the  name  3  with  the  phrase  'an 
integer  between  4  and  5  ' :  in  the  sense  in  which  we 
can  significantly  assert  that  3  is,  we  may  assert  that  an 
integer  between  4  and  5  zs  not ;  in  other  words,  no  true 
character  can  be  assigned  to  this  proposed  subject, 
except  what  is  involved  in  our  understanding  of  its 
meaning,  namely  that  it  belongs  to  the  general  category 
of  integer,  and  that  it  is  to  be  greater  than  4  and  less 
than  5.  Generalising  from  this  example,  it  will  be  seen 
that  such  a  subject-term  is  defined  first  by  reference  to 
a  general  category  (in  the  above  case  that  of  number) 
and   next,   by  a  proposed  means   of  determining  or 


selecting  out  of  the  members  of  that  category,  a  par- 
ticular example  related  in  a  defined  way  to  other  things. 

With  reference  to  the  category  to  which  the  subject- 
term  S  by  definition  belongs,  any  difference  of  category 
is  naturally  associated  with  an  apparent  difference  in 
the  meaning  of  'is.'  In  particular  there  is  a  range  of 
subjects  for  which  the  word  *  exists  '  would  be  naturally 
substituted  for  *is.'  Thus  it  may  be  agreed  that  what 
is  manifested  in  space  and  time  may  be  said  to  exist: 
hence  we  raise  such  questions  as  whether  God  exists, 
or  whether  the  centaur  Cheiron  existed,  or  whether 
William  Tell  existed.  The  objects  intended  to  be 
denoted  by  these  subject-terms  may  be  said  to  belong 
to  the  category  of  the  existent  whether  the  propositions 
asserting  their  existence  are  true  or  false.  Thus  we 
must  maintain,  in  accordance  with  the  nature  of  the 
definition  of  God,  that  *  God  is  an  existent,' — this  being 
a  merely  verbal  or  analytic  proposition ;  but  the  question 
of  the  truth  of  the  synthetic  or  real  proposition  '  God 
exists '  remains  problematic.  The  same  holds  of  Cheiron 
and  William  Tell.  On  the  other  hand  what  is  denoted 
by  such  a  subject-term  as  3  or  an  integer  between  4 
and  5  would  not  be  called  an  existent.  Thus  we  main- 
tain that  there  is  no  difference  in  the  force  of  the  word 
*is'  in  its  isolated  usage;  but  that  if  any  difference 
appears — as  when  we  substitute  *  exists  '  for  '  is ' — this 
is  merely  due  to  a  difference  in  the  category  of  the 
subject-term,  w^hich  again  presupposes  a  difference  in 
the  types  of  adjectives  that  are  properly  predicable  of  it. 

When  the  proposition  '  5  is  '  is  under  consideration  ' 
it  must  be  understood  that  the  term  S  is  not  an  ordinary 
singular  name  but  one  of  a  peculiar  nature  that  has  not, 


76 


CHAPTER  V 


NEGATION 


17 


i  I  think,  been  recognised  by  logicians.     Such  a  name 
will  be  designated  by  the  prefix  *a  certain.'    Consider 
the  following  propositions :  *  A  certain  man  was  both  a 
philosopher  and  a  historian/  'a  certain  integer  between 
3  and  1 1  is  prime,'  *  a  certain  novel  has  no  hero,'  *  a 
certain  flash  of  lightning  was  vivid.'    The  truth  or  falsity 
of  these  propositions  could   only  be  decided  by  the 
hearer  if  for  the  phrase  '  a  certain  '  is  substituted  *  some 
or  other,'  '  one  or  more,'  so  that  for  him  the  reference 
is  indeterminate.    Thus,  of  Hume  and  of  Xenophon  it 
is  true  that  they  were  both  historians  and  philosophers ; 
between  3  and  1 1  there  are  two  numbers — 5  and  7 — 
that  are  prime;  and  the  other  examples  are  equally 
ambiguous.    We  must  suppose  that  the  speaker  has 
in  mind   a   single   determinate   philosopher-historian, 
number,  novel  or  flash,  which  has  been  identified  by  him, 
and  to  which  therefore  he  may  return  in  thought.   From 
these  examples  we  see  that  a  term  may  be  properly  called 
uniquely   singular  for  the  asserter,   although   in   fact 
there  may  be  several  objects  answering  to  its  explicit 
description.     Thus  from  a  proposition  with   the  pre- 
designation  *  a  certain'  maybe  inferred  the  corresponding 
proposition  with  the  predesignation  *some  or  other,' 
though  of  course  not  conversely.     This  points  to  two 
modes  in  which  what  is  technically  called  the  particular 
I  proposition  can  be  inferred:  first,  from  premisses  one 
;i.  of  which  is  itself  particular ;  and  secondly,  from  a  specific 
instance  for  which  the  predesignation  '  a  certain  '  stands. 
Now  it  is  the  latter  form  of  proposition  which  raises 
the  problem  of  the  significance  of  the  proposition  *5  is.' 
For  the  asserter,  the  contradictory  of  the  proposition 
that  *  A  certain  man  was  both  an  historian  and  a  philo- 


sopher '  would  be  that  the  person  of  whom  he  is  thinking 
was  not  both  an  historian  and  a  philosopher :  or  the 
contradictory  of  the  proposition  that  a  certain  integer 
between  3  and  1 1  is  prime,  would  be  that  that  same 
integer  is  composite  ;  whereas  for  the  hearer,  who  can 
only  understand  the  given  propositions  as  particular, 
the  contradictory  in  the  first  case  would  be:  'No  man 
is  both  an  historian  and  a  philosopher'  and  in  the  second, 
*  No  integer  between  3  and  11  is  prime.'  In  fact,  in 
denying  the  proposition  that  a  certain  integer  between 
3  and  1 1  is  prime,  we  must  mentally  specify  the  integer 
about  which  we  are  thinking,  and  assert  that  this  integer 
is  composite.  The  form  of  the  statement  thus  reached 
is  equivalent  to  that  of  the  conclusion  in  the  process  of 
obversion,  but  it  is  not  obtained  here  (as  in  obversion) 
by  the  medium  of  the  purely  negative  premiss,  but 
directly  by  mentally  specifying  the  number  under  con- 
sideration. 

§  5.  It  remains  to  explain  more  precisely  the  nature 
of  the  denial  of  '  S\^  P'  which  combined  w^ith  *  5  is ' 
yields  the  conclusion  'S  is  non-P!  The  proposition 
which  merely  denies  'S  \s  P'  must  be  understood  to  in- 
volve a  hypothetical  element.  Consider,  for  example, 
the  statement  *  Anyone  who  calls  this  afternoon  is  not 
to  be  admitted '  ;  this  proposition  does  not  contain  any 
categorical  assumption  that  somebody  will  call,  and  may 
be  otherwise  expressed  in  an  explicitly  hypothetical 
form  *  If  anyone  calls  he  is  not  to  be  admitted.'  Com- 
bining this  premiss  with  the  further  ascertainable  fact 
that  a  certain  person  has  called,  the  obvious  conclusion, 
that  this  person  is  not  to  be  admitted,  follows.  In  general 
the  symbols  that  we  have  used,  namely  that  '  S  \s  P^ 


7^ 


CHAPTER  V 


NEGATION 


79 


is  false  and  that  *  5  is,'  may  be  explained  by  making 
explicit  the  descriptive,  adjectival,  or  connotative  factor 
in  the  term  symbolised  by  5,  which  factor  we  shall 
symbolise  by  M,    The  negative  premiss  then  becomes : 
•If  anything  is  M  it  will  not  be  P' \  the  categorical 
premiss  becomes:    'A  certain  thing  is  J/.'     In  this 
formulation  the  symbol  5  does  not  appear.     Now  5 
stands  for  a  certain  thing  which   has  not   yet   been 
identified  and  which  is  only  presented  in  thought  by 
the  general  description  M ;  or  briefly  S  is  to  mean  '  a 
certain  thing  which  is  ^.'     In  transforming  the  pro- 
position 'Anything  that  is  M  will  not  be  P'  into  the 
form  *  5  will  not  be  /^ '  we  introduce  the  factor  '  a  cer- 
tain thing.'    And  in  transforming  the  proposition  *A 
certain  given  thing  is  M'  into  the  form  * 5  is'  we  have 
transferred  the  whole  of  the  adjectival  component  in  the 
proposition  from  the  predicate  to  the  subject:  or  other- 
wise, the  two  propositions  maybe  rendered  'Anything 
that  may  be  given  having  the  character  M  will  not  be 
P '  and  '  A  certain  thing  having  the  character  M  is 
given.'    Thus  it  is  not  strictly  correct  to  use  the  same 
symbol    S   in   our   two   propositions,   since   the   only 
differentiating  element  of  meaning  in  the  term  S  in  the 
negative  premiss  is  the  adjectival  or  descriptive  com- 
ponent, whereas  in  the  categorical  premiss  the  substan- 
tival component  enters  along  with  the  adjectival.    It 
follows  that  the  analysis  given  is  not  restricted  to  the 
negative  form  of  our  first  premiss,  since  the  same  kind 
of  syllogism  would  apply  to  an  affirmative  conclusion  : 
the  essential  characteristic  of  the  first  premiss  is  its 
hypothetical  character,  as  opposed  to  the  other  premiss 
which  is  categorical.    Thus  the  affirmative  case  would 


be  rendered  'Anything  that  may  be  given  having 
the  character  M  will  be  /*,'  '  A  certain  thing  with  the 
character  M  is  given'  therefore  'This  thing  having  the 
character  M  will  be  P,' 

§  6.  In  the  course  of  this  final  explanation  it  will  be 
noted  that  for  the  formula  '6^  is'  in  which  no  determinate 
adjective  is  predicated,  we  might  substitute  'S  is  given' 
or  '5  is  real.'  Now  the  words  'given '  and  '  real '  though 
of  course  grammatically  adjectival,  are  not  in  the  logical 
sense  adjectival,  for  their  meaning  does  not  contain  any 
indication  of  character  or  relation.  It  may  be  remarked 
in  passing  that  the  application  of  the  term  'real'  includes 
but  goes  beyond  that  of  the  word  'given.'  The  postulate 
that  has  to  be  assumed  is  that,  however  indeterminately 
we  may  have  been  able  to  characterise  it,  the  real  must 
have  some  determinate  character.  We  thus  return  to 
our  first  exposition  of  the  force  of  the  incomplete  pre- 
dication '  5  is  ' :  namely  that  S,  as  being  real,  must  have 
some  determinate  character  although  it  may  be  that 
this  character  cannot  be  completely  or  exactly  known 
by  any  finite  intelligence. 


8o 


THE  PROPER  NAME  AND  THE  ARTICLES    8i 


p.  96 


CHAPTER  VI 

THE  PROPER  NAME  AND  THE  ARTICLES 

§  I.  A  simple  proposition  ' S\s>  P'  involves  as  sub- 
ject a  single  uniquely  determined  substantive,  and  as 
predicate  a  single  uncompounded  adjective;  a  singular 
proposition  must  satisfy  the  first  of  these  conditions, 
but  its  predicate  may  be  simple  or  compound.  Pro- 
positions of  this  nature  give  rise  to  the  question  how 
the  reference  in  the  subject  can  be  uniquely  determined. 
Speaking  generally,  singular  names  may  be  divided  into 
two  classes  according  as  they  contain  or  do  not  contain 

I  an  explicit  adjectival  or  relational  component :  to  the 
first  class  belong  such  terms  as  'the  smallest  planet,' 
*the  king  of  England  who  signed  Magna  Charta,'  'the 
cube  root  of  8  ' ;  to  the  second  class  '  Mercury,'  *  John,' 
and  '  2 ' ;  and  the  former  will,  for  convenience,  be  referred 
to  as  descriptive  or  significant,  the  latter  as  proper  or 
non-significant. 

Compare  now  the  proper  name  'Poincar^'  with  the 
descriptive  name  'the  President  of  France.'  In  order 
that  this  latter  term  may  have  unique  application  its 
component '  France'  must  have  unique  application  :  and 
hence  here,  as  in  almost  every  case,  the  uniqueness  of 
a  descriptive  name  is  only  secured  through  its  reference 

I  to  a  proper  name.  On  the  other  hand  we  shall  find  that 
many  so-called  proper  names  contain  a  descriptive  factor: 
thus  the  term  'England'  contains  the  termination  'land' 
which  would  be  normally  understood  as  bringing  the 


term  '  England '  within  the  general  class  expressed  by 
the  word  'land.'  This,  however,  would  not  explicitly 
hold  of  the  name  '  France ' ;  but,  in  point  of  fact,  just 
as  England  might  be  taken  to  mean  'the  land  of  the 
Angles,'  so  might  France  be  taken  to  mean  'the  land 
of  the  Franks.'  In  the  same  way  the  terms  '  Mr 
Gladstone '  and  '  Lord  Beaconsfield '  have  the  partial 
significance  expressed  by  the  prefixes  'Mr'  and  'Lord' 
respectively.  As  another  example,  the  name  '  Mont 
Blanc,'  though  etymologically  equivalent  to  'white 
mountain '  and  therefore  apparently  completely  signifi- 
cant, must  yet  be  called  a  proper  name  since  its  appli- 
cation is  not  to  any  white  mountain,  but  to  a  specific  one.  , 
We  are  here  in  the  reverse  position  to  that  reached  in 
discussing  the  term  'England';  for  in  'England' we 
detected  the  concealed  element  of  significance  indicated 
by  the  termination  'land,'  while  in  'Mont  Blanc'  we 
have  detected  the  concealed  element  of  non-significance 
which  prevents  us  from  applying  the  name  to  any  white 
mountain  indiscriminately.  Now,  attributing  to  the  term 
*  Mont  Blanc '  the  maximum  of  significance  that  it  can 
bear,  and  agreeing  that  there  would  be  a  species  of  in- 
correctness in  using  the  term  for  any  object  which  had 
not  the  characteristics  'white '  and  '  mountain,'  yet  this 
admitted  significance  is  not  the  sufficient  ground  for 
applying  the  term  as  it  is  understood  by  those  who  use 
it  with  a  common  agreement  as  to  its  unique  applica- 
tion. We  may  therefore  say  that  any  name  which  is 
commonly  called  a  proper  name  has,  so  far  as  our 
analysis  has  proceeded,  a  residual  element  of  non-signifi- 
cance over  and  above  such  significance  as  is  naturally 
recognised  in  the  verbal  structure  of  the  name. 

J.  L.  6 


82 


CHAPTER  VI 


§  2.    But  if  we  allow  of  any  name  that  it  contains  an 
element  of  non-significance,  how  is  it  possible  that  this 
name  should  be  understood  as  applying  to  the  same 
object  when  used  at  different  times  or  by  different 
persons  or  in  different  and  varying  connections  ?  Where 
the  name  denotes  a  substantive,  the  possibility  that  it 
should  mean  the  same  substantive  when  used  in  different 
propositions,   involves  the  possibility  of  substantiyal 
identification.    A  similar  process,  i.e.  adjectival  identifi- 
cation, is  involved  when  the  same  adjective  is  used  in 
different  connections.    Whether  we  ask  how  the  sub- 
stantive-name '  Snowdon,'  for  instance,  can  be  under- 
stood to  stand  for  a  definite  substantive,  or  how  the 
adjective-name  *  orange'  can  be  understood  to  stand  for 
a  definite  adjective,  we  are  in  fact  confronted  with  pre- 
cisely the  same  logical  problem ;  and  hence,  if  we  regard 
the  name  'Snowdon'  as  a  proper  substantive-name,  we 
must  regard  ^orange'  as  a  proper  adjective-name.   The 
analogy  may  be  pressed  a  little  further  and  applied  to 
complex  names;  for  just  as  an  adjective  name  is  ex- 
hibited as  significant  when  it  is  expressed  in  the  form 
of  adjectives  combined  in  certain  relations,  so  a  sub- 
tantive   name   is   exhibited   as  significant  when  it  is 
'  expressed  in  the  form  of  substantives  combined  in  cer- 
tain relations.    Note  the  analogy,  for  example,  between 
the  complex  adjective-name  'the  colour  between  red 
and  yellow'  and  the  complex  substantive-name  'the 
highest  mountain  in  Wales.'    Here  the  former  involves 
the  proper  adjective-names  '  red '  and  '  yellow,'  just  as 
the  latter  involves  the  proper  substantive-name  'Wales.' 
There  is,  then,  in  every  explication  of  significance,  a 
residual  element  in  which  we  reach  either  a  substantive 


THE  PROPER  NAME  AND  THE  ARTICLES         83 

name  or  an  adjective-name  which  can  no  longer  be  de- 
fined in  this  form  of  analysis.  And  further,  the  explication 
of  the  significance  of  this  residual  proper  substantive  or 
proper  adjective-name  involves  the  conception  of  iden- 
tity: in  the  one  case,  substantival  identity — which  is 
implied  when  we  understand  that  the  substantive  de- 
noted by  the  word  '  Snowdon '  in  one  proposition  is 
identical  with  that  denoted  by  *  Snowdon '  in  another 
proposition;  and,  in  the  other  case,  adjectival  identity — 
which  is  implied  when  we  understand  that  the  adjective 
denoted  by  'orange'  in  one  proposition  is  identical  with 
the  adjective  denoted  by  'orange'  in  another  proposition. 
Our  first  approximate  account  of  a  proper  name  is 
then,  that  the  intended  application  of  the  given  name 
is  to  an  object — whether  it  be  substantive  or  adjective — 
which  is  identical  with  the  object  to  which  it  may  have 
been  previously  understood  as  applying  in  another  pro- 
position.    For  example,  to  explain  what   I   mean  by 

orange'  I  could  say:  'You  understand  the  word  colour: 
and  I  shall  mean  by  "orange"  the  colour  which  you  can 
discern  as  characterising  the  object  to  which  I  am 
pointing.  And  when  you  identify  the  colour  of  any  ob- 
ject with  the  colour  of  this,  its  colour  is  to  be  called 

orange".'  The  possibility  of  such  appeal  presupposes 
that  colour  can  be  perceptually  identified  in  different 
objects,  apart  from  any  other  agreements  or  differences 
that  the  objects  may  manifest.  In  the  same  way  the 
explication  of  a  proper  substantive-name  requires  a 
similar  appeal,  which  assumes  the  possibility  of  identi- 
fying a  concrete  object  when  it  may  be  presented  or 
thought  about  in  different  contexts.  Thus,  if  it  was 
asked  whom  I  meant  when  I  talked  of  Mr  Smith,  I 

6-2 


84 


CHAPTER  VI 


THE  PROPER  NAME  AND  THE  ARTICLES 


85 


might  say:  *  1  mean  the  man  to  whom  you  were  mtro- 
duced  yesterday  in  my  study.'  The  agreement,  there- 
fore, which  can  be  maintained  in  the  apphcation  of  a 
proper  name  amongst  those  who  continue  to  use  it  with 
mutual  understanding,  is  secured  by  what  in  a  quite 
general  way  we  may  call  the  metjiod^  djjiHoduct^^ 
An  object  is  introduced,  and  in  the  introduction  a  name 
is  given,  and  when  further  reference  is  intended  to  the 
same  object,  the  name  is  repeated  which  was  given  in 
the  act  of  introduction.  In  this  way  the  nature  of  the 
residual  element  or  undefinable  factor  in  adjectives  and 
relations  as  well  as  that  in  the  case  of  proper  names  is 

further  explained.  . 

^        §  3.    It  is  worth  pausing  here  to  point  out  a  contusion 
frequently  made  in  discussing  the  nature  of  the  proper 
name.    The  confusion  is  that  between  the  cause  which 
has  led  people  to  choose  one  name  rather  than  another 
name  for  a  given  application,  with  the  reason  for  ap- 
plying the  name— once  chosen— to  one  object  rather 
than  to  another  objed.    This  confusion  again  can  be 
paralleled  in  ordinary  adjectival  names  as  well  as  in 
substantival  names :  thus,  it  is  one  thing  to  assign  the 
etymological  causes  of  the  use  of  the  name  '  indigo 
rather  than  some  other  name  to  denote  a  particular 
colour,  and  another  thing  to  assign  the  reason  for  ap- 
plying the  name,  when  it  has  once  come  into  common 
usage,  to  one  of  the  colours  rather  than  to  some  other: 
the  reason  for  this  latter  is  that  the  colour  presented  in 
a  Riven  instance  is  identical  with  that  to  which  the  name 
indigo  was  originally  given.    This  is  exactly  parallel  to 
the  ground  on  which  we  should  justify  our  applying  the 
name  *  Roger  Tichborne'  to  the  man  presented  in  court: 


namely,  the  presumed  identity  of  the  man  before  us 
with  the  man  to  whom  his  godparents  had  given  the 
name.  Why  they  chose  that  name  rather  than  some 
other  name  is  a  matter  for  historical  enquiry.  But,  to 
repeat,  the  etymological  or  historical  account  of  a  name 
must  not  for  a  moment  be  confused  with  its  significance, 
or  with  what,  in  the  case  of  proper  names  (substantival 
or  adjectival),  takes  the  place  of  significance  as  the 
condition  of  mutual  understanding. 

§  4.  This  discussion  is  closely  bound  up  with  the 
different  ways  in  which  the  articles,  indefinite  and  de- 
finite, are  used.  The  indefinite  article  in  its  most  general 
and  completely  indeterminate  meaning  is  illustrated  by 
such  assertions  as  *  A  man  must  have  been  in  this  room,' 
*  We  need  a  sweep,'  *  You  ought  to  make  a  move  with 
your  bishop.'  Now  if  we  compare  this  use  of  the  article 
with  its  meaning  when  it  occurs  at  the  beginning  of  a 
narrative  as  for  instance:  'Once  upon  a  time  there  was 
la  boy  who  bought  a  beanstalk,'  we  note  an  important 
difference  in  its  significance.  I  n  the  first  set  of  examples 
the  full  significance  of  the  article  is  made  explicit  by 
substituting  'some  or  other':  e.g.  '  Some  or  other  man 
must  have  been  in  this  room' ;  in  the  case  of  a  narrative, 
where  the  article  prepares  the  way  for  future  references 
to  period,  person  or  place,  it  means — not  'some  or 
other' — but  'a  certain.'  Indeed  our  story  might  more 
logically  have  begun  '  At  a  certain  time  a  certain  boy 
bought  a  beanstalk.'  When  the  indefinite  article  is  used 
in  this  way  to  introduce  some  period,  person  or  place 
not  otherwise  indicated,  it  will  henceforward  be  called 
|the  Introductory  Indefinite,  to  distinguish  it  from  the 
\lternative  Indefinite. 


86 


CHAPTER  VI 


THE  PROPER  NAME  AND  THE  ARTICLES 


87 


Now  suppose  the  narrative  to  continue:  *This  boy 
was  very  lazy';  the  phrase  'this  boy'  means  *the  boy 
just  mentioned,'  the  same  boy  as  was  introduced  \.o  us 
by  means  of  the  indefinite  article.    H  ere  the  article  *  this, ' 
or  the  analogous  article  'the,'  is  used  in  what  may  be 
called  its  referential  sense.    The  linguistic  condition 
necessary  to  render  such  reference  definite  is  that  only 
one  object  of  the  class  (whether  person,  period,  or  place) 
should  have  been  immediately  before  mentioned.    Other 
variations  of  the  Referential  Definite  are  such  phrases  as 
'  the  former'  and  'the  latter,' which  may  be  required  to  se- 
curedefinite  reference.  The  above  analysisbrings  out  the 
necessarily  mutual  association  of  the  introductory  use  of 
xh&indefinite  article  with  th^re/erential use  oi  the  definite 
article.  Again,  instead  of  beginning  the  second  sentence 
with  the  phrase  'this  boy,'  language  permits  us  to  use 
a  pronoun:  thus  the  word  'he,'  in  general,  is  sufficient 
to  denote  a  specific  individual  understood  by  the  verbal 
context;  so  that  here  the  pronoun  serves  precisely  the 
same  logical  function  as  the  referential  definite  article 
'the'  or  'this.'    A  still  more  important  further  develop- 
ment of  the  referential  'the'  comes  up  for  consideration 
when,  instead  of  depending  upon  immediacy  of  con- 
text—as in  the  preceding  cases  of  'this'  and  'he' — we 
refer  to  an  historical  personage  who  has  a  wide  circle  of 
acquaintance  as  (e.g.)  'The  well-known  sceptical  phi- 
losopher of  the  eighteenth  century.'     Here  the  phrase 
•the  well-known'  functions  as   a   referential   definite, 
though  there  may  have  been  no  immediately  previous! 
mention  of  Hume,  it  being  assumed  that  a  certain  phi- 
losopher will  be  unambiguously  suggested  to  readers 
in  general,  in  spite  of  the  fact  that  there  may  have  been 


more  than  one  person  answering  to  the  description  'scep- 
tical philosopher  of  the  eighteenth  century.'    This  ex- 
tended use  of  the  referential  definite  is  quite  interestingly 
illustrated  in  Greek,  where  a  proper  name  is  prefixed 
by  the  definite  article  '  6 ' ;  a  usage  which  appears  very 
happily  to  bring  out  the  precise  function  of  the  proper 
name,  as  referring  back  to  an  individual  who  was  origin- 
ally introduced  in  history  or  otherwise  under  that  name. 
The  same  holds   in   English  of  geographical  proper 
names,  e.g.  the  Thames,  the  Hellespont,  the  Alps,  the 
Isle  of  Wight,  etc.     Lastly,  in  a  narrative,  the  juxta- 
position of  a  proper  name  with  the  introductory  in- 
definite supplies  a  substitute  for  the  referential  definite. 
Thus  our  story  about  the  beanstalk  which  begins  with 
the  introductory  indefinite  'a  boy'  may  be  continued 
either  by  using  the  phrase  'this  boy' — involving  the 
referential  article — or  by  the  pronoun  'he';  or  thirdly 
by  the  proper  name  which  prepares  the  way  for  repeated 
reference  to  the  same  boy:  'Once  upon  a  time  there 
was  a  boy  named  Jack  who  bought  a  beanstalk.'   It  will 
,  be  noted  therefore  that  the  way  in  which  the  proper 
name  occurs  in  a  narrative  where  it  secures  continuity 
of  reference,  illustrates  the  same  principle  as   its  use 
in  ordinary  intercourse,   where  it  ensures  agreement 
amongst  different  persons  as  to  its  single  definite  appli- 
i  cation :  in  both  cases,  the  understanding  of  the  applica- 
tion of  the  name  involves  reference  back  to  the  act  of 
introduction,  when  the  name  was  originally  imposed. 

There  is  an  important  analogy  between  the  singular 
descriptive  name  of  the  kind  illustrated  by  'the  well- 
known  sceptical  philosopher  of  the  eighteenth  century' 
and  the  proper  name,  in  that  frequently  it  is  only  within 


SB 


CHAPTER  VI 


THE  PROPER  NAME  AND  THE  ARTICLES 


89 


a  narrower  or  wider  range  of  context  that  the  proper 
name  may  be  said  to  have  a  uniquely  determined  appli- 
cation. Thus,  within  a  family,  the  name  'John'  maybe 
understood  to  denote  one  brother  of  that  name ;  whereas, 
in  a  certain  period  in  English  history,  it  will  denote  the 
king  who  signed  Magna  Charta.  That  uniqueness  of 
reference  is  relative  to  a  particular  context  is  similarly 
seen  in  such  phrases  as  'the  table,'  'the  garden,'  *the 
river,'  which  though  applicable  to  different  objects  in 
different  contexts  are  understood  within  a  given  circle 
or  in  a  given  situation  to  have  a  uniquely  determined 
application.  The  article  'the'  used  in  such  cases  may 
be  called  Indefinite  Definite,  to  distinguish  it  from  the 
i  most  definite  of  all  uses  of  the  article,  namely  where 
the  unique  application  is  understood  without  any  limita- 
tion of  context — in  cases,  for  example,  like  'the  sun,' 

*the  earth.' 

We  have  thus  divided  articles  (and  what  are  logic- 
ally equivalent  to  articles)  into  four  classes:  (i)  the 
Indefinite  Indefinite,  otherwise  the  Alternative  Indefi- 
nite; (2)  the  Definite  Indefinite,  otherwise  the  Instan- 
tial  Indefinite,  best  expressed  by  the  phrase  'a  certain,' 
which  includes  the  Introductory  Indefinite;  (3)  the 
Indefinite  Definite,  otherwise  the  Contextual  Definite, 
which  includes  the  Referential  Definite;  and  (4)  the 
Definite  Definite,  for  which  the  understood  reference 
is  independent  of  context. 

§  5.  A  special  form  of  the  contextual  definite  which 
is  to  be  distinguished  from  the  referential,  is  expressed 
by  the  terms  'this'  and  'that'  when  used  as  demonstra- 
tives. Literally,  the  demonstrative  method  is  limited 
to  the  act  of  introducing  an  object  within  the  scope  of 


perception.  But,  when  we  point  with  the  finger,  for 
instance,  to  a  particular  person  or  mountain  or  star,  our 
attempt  to  direct  the  attention  of  the  hearer  to  the  object 
intended  may  or  may  not  succeed :  if  successful,  it  will 
be  because  there  is  no  other  conspicuous  object  belong- 
ing to  the  class  indicated  by  the  use  of  the  general 
significant  name  (person,  mountain,  star,  as  the  case 
may  be)  within  the  range  of  space  to  which  we  have 
directed  attention.  The  condition  for  securing  unam- 
biguity  is  not  that  there  should  be  only  one  object  of 
the  specified  class  within  the  range  indicated,  but  that 
there  should  be  only  one  such  visible  object ;  and  here 
observe  a  parallel  between  the  demonstrative  definite, 
and  the  case  illustrated  by  the  example  'the  well-known 
sceptical  philosopher  of  the  eighteenth  century.' 

§  6.  At  this  point  in  our  discussion  let  us  consider 
the  special  difficulty  which  attaches  to  the  notion  of  a 
proper  name.  This  problem  presents  a  dilemma.  If 
we  maintain  that  the  proper  name  is  non-significant  in 
some  sense,  then  it  would  follow  that  any  propositional 
phrase  that  might  contain  the  proper  name  would  be 
non-significant  in  the  same  sense.  If,  on  the  other  hand, 
we  attempt  to  assign  some  definite  significance  to  the 
proper  name,  this  will  entail  our  substituting  a  uniquely 
descriptive  name  as  equivalent  in  meaning  to  the  proper 
name,  in  which  case  the  distinction  between  the  de- 
scriptive name  and  the  proper  name  would  vanish. 

This  problem  raises  a  question  relating  to  the  wider 
problem  of  the  definition  of  words  or  phrases.  Taking 
the  two  words  'valour'  and  'courage,'  the  brief  formula 
'valour  means  courage'  is  seen  on  reflection  to  be  im- 
perfectly expressed.    Everybody  would  agree  that  what 


k 


90 


CHAPTER  VI 


is  intended  here  is  that  the  two  terms  valour  and  courage 

have  the  same  meaning;  i.e.  that  the  quality  meant  by 

the  one  term  is  the  same  as  the  quality  meant  by  the 

other.    Hence  a  more  correct  expression  than  'valour 

means  couragfe'  would  be  'the  word  valour  means  what 

is  meant  by  the  word  courage.'    Where  a  phrase  m- 

stead  of  a  single  word  is  under  consideration  the  same 

principle  is  involved.    For  example,  */  is  a  factor  of  q 

means-what-is-meant-by  'q  is  divisible  by  /';  or  again 

'some  benefactor  of  A'  means-what-is-meant-by  'one 

or  other  person  who  has  benefited  A!    These  illustra- 

'  tions  bring  out  the  distinction  between  [a)  the  relation 

\  which  one  word  or  phrase  may  bear  to  another  word  or 

.  phrase,  and  {b)  the  relation  which  a  word  or  phrase  may 

i  bear  to  what  is  called  its  'meaning.' 

Now  the  propositions  which  allow  us  to  substitute 
one  phrase  for  another  may  be  called  bi-verbal  defi- 
nitions'; and  the  relation  that  is  to  be  affirmed  as  hold- 
ing between  two  such  phrases  must  be  expressed  in 
the  complex  form  'means  what  is  meant  by,'  or  even — 
when  we  distinguish  between  the  phrase  which  has  not 
been  understood  and  that  which  has  been  understood— 
in  the  still  more  complicated  form  'is  to  be  understood 
to  mean  what  has  been  understood  to  be  meant  by.' 
This  last  complication  brings  out  the  purpose  that  a 
i definition  has  always  to  serve;  namely  the  elucidation 
of  a  phrase  assumed  to  require  explanation  in  terms  of 
a  phrase  presumed  to  be  understood. 

^  It  has  been  suggested  that  a  more  correct  substitute  for  ^bi- 
verbal  definition'  wo.uld  be  'translation:  But  whichever  terminology 
is  employed,  the  distinction  between  the  kind  of  definition  called 
translation  and  some  more  ultimate  definition  remains. 


THE  PROPER  NAME  AND  THE  ARTICLES    91 

This  formulation  of  the  bi-verbal  definition  leads  us 
to  consider  what,  in  contrast,  we   shall  call   the  uni- 
verbal  definition.   When  we  speak  of  a  phrase  as  being 
'already  understood,'  it  is  equivalent  to  saying  that  the 
meaning  of  the  phrase  is  known.     The  formula  that 
'phrase  p  means  what  is  meant  by  phrase  q^  in  short, 
raises  the  question,  What  is  it  that  phrase  q  means? 
Let  us  first  consider  the  kind  of  entity  that  a  phrase 
could  mean.    Phrases  which  would  be  grammatically  or 
logically  distinguished  in  regard  to  type  or  category 
would  mean  entities  belonging  to  correspondingly  dif- 
ferent types  or  categories :  thus  one  phrase  would  mean 
a  certain  proposition;  another  would  mean  a  certain 
adjective,  another  a  certain  substantive,   and    so  on. 
Thus  'courage'  means  a  certain  adjective  or  quality-of- 
conduct,  'horse'  means  a  certain  substantive  or  kind-of- 
animal.    A  phrase  prefixed  by  an  article  such  as  a,  the, 
some,  every,  any,  requires  special  consideration.    Thus, 
if  we  were  to  substitute  for  such  phrase  any  phrase  that 
means  what  is  meant  by  the  given  phrase,  the  article 
or  some  equivalent  would  still  remain.    Thus  'the  first 
novel  of  which  Scott  was  the  author'  means  what  is 
meant  by  'the  romance  that  was  written  by  Scott  before 
any  other  of  his  romances.'    In  this  bi-verbal  substitu- 
tion the  word  'the'  is  retained.     Now  consider,  in  con- 
trast to  the  proposition  stating  the  equivalence  in  mean- 
ing of  the  above  phrases,  the  proposition  '  The  first 
novel  written  by  Scott  was  called  Waverley\  or,  inas- 
much as  there  is  only  one  novel  that  is  known  bearing 
this  name,   we   may  put  the  statement   in  the  form: 
*The  first  novel  written  by  Scott  was  the  novel  called 
Waver  ley'   Such  a  proposition  is  of  nearly  the  same 


92 


CHAPTER  VI 


THE  PROPER  NAME  AND  THE  ARTICLES 


93 


type  as  *The  author  of  Waver  ley  was  the  author  of 
Marmion'     In  both  of  these  propositions  the  relation 
of  identity  is  asserted  in  regard  to  two  uniquely  de- 
scriptive terms.    But  neither  of  these  propositions  is 
verbal ;  in  neither  case  could  we  substitute  for  the  rela- 
tion of  identity  the  expression  'means  what  is  meant  by.' 
Hence  we  are  not  identifying  the  meaning  of  the  two 
phrases:  i.e.  we  are  not  identifying  what  is  meant  by 
one  phrase  with  what  is  meant  by  the  other.    What 
then  is  it  that  we  are  identifying.^     In  the  language  of 
Mill  we  should  say  we  are  identifying  what  is  denoted — 
and  in  the  language  of  Frege  what  is  indicated  or  (as 
we  prefer  to  say)  factually  indicated — by  the  one  phrase 
with  what  is  denoted  or  factually  indicated  by  the  other. 
Now,  as  our  term  suggests,  an  appeal  X.o  fact  is  required 
in  order  to  understand  what  it  is  that  is  factually  indi- 
cated in  distinction  from  what  is  meant  by  a  certain 
phrase.      Hence,   though  a  knowledge   of  the   usage 
of  language  alone  is  sufficient  to  know  what  a  phrase 
means,   a  knowledge   of  something  more  than  mere 
linguistic  usage  is  required  to  know  what  a  phrase  de- 
notes or  factually  indicates,  whenever  we  are  dealing 
with  a  phrase  that  indicates  something  different  from 
what  it  means.    The  word  'courage'  or  the  phrase  'not 
flinching  from  danger'  is  of  such  a  nature  that  there  is 
no  distinction  between  what  it  means  and  what  it  indi- 
cates or  denotes;  it  is  only  phrases  prefixed  by  an  article 
or  similar  term  for  which  the  distinction  between  mean- 
ing and  indication  arises.    Turn  now  to  the  peculiarities 
of  the  illustration  given  above:  'The  first  novel  written 
by  Scott  was  the  novel  called  Waverley!    The  inter- 
pretation of  this  statement  is  that  the  object  indicated 


by  the  phrase  that  stands  first  is  the  same  as  that 
indicated  by  the  phrase  'the  novel  called  Waverley 
although  the  meanings  of  the  two  phrases  differ.  Take 
a  parallel  case:  'The  colour  of  the  object  at  which  I 
am  pointing  is  identical  with  the  colour  that  is  called 
red ' ;  here  again  the  identity  of  what  is  indicated  by  the 
two  phrases  does  not  carry  with  it  identity  in  what  is 
meant  by  the  two  phrases.  In  short,  where  we  have 
an  identification  of  what  is  indicated  in  spite  of  non- 
identity  in  what  is  meant,  we  recognise  that  the  state-  , 
ment  of  identity  is  not  merely  verbal  but  factual.  « 

In  the  above  illustrations  we  have  taken  such  names 
as  Scott  and  Waverley  to  exemplify  names  universally 
recognised  as  proper;  while  the  phrase  'the  first  novel 
written  by  Scott' — or  any  phrase  having  the  same  mean- 
ing—would be  called  descriptive  in  a  sense  primarily 
intended  as  antithetical  to  proper.  Now  one  step  was 
taken  to  bridge  this  antithesis  when  we  used  the  proper 
name  in  the  extended  phrase  'the  novel  called  Waver- 
ley \  i.e.  the  single  name  Waverley  is  a  proper  name 
and  the  compound  phrase  'the  novel  called  Waverley' 
is  constructed  in  the  form  of  a  descriptive  name.  We 
are  thus  leading  up  to  the  view  that  what  is  indicated 
by  the  descriptive  phrase — *the  novel  called  Waverley  — 
is  identical  with  what  is  meant  by  the  proper  name 
'Waverley.'  Thus,  in  interpreting  the  simple  propo- 
sition '  Waverley  was  the  first  novel  written  by  Scott,' 
which  is  recognised  at  once  to  be  factual  not  verbal,  we 
are  identifying  what  is  factually  indicated  by  the  subject 
and  predicate  terms  respectively ;  and  in  the  case  of  the 
proper  name  'Waverley,'  what  it  factually  indicates  is 
indistinguishable  from  what  it  means.    Hence  it  seems 


94 


CHAPTER  VI 


THE  PROPER  NAME  AND  THE  ARTICLES 


95 


legitimate  or  possible  to  define  a  proper  name  as  a  name 
which  means  the  same  as  what  it  factually  indicates, 

87.   We  may  now  introduce  the  technical  term '  osten- 
sive'  which  will  suggest  as  its  opposite  the  familiar  term 
'intensive.'    A  proper  name  may  be  said  to  be  osten- 
sively  definable  in  contrast  to  those  more  ordinary  terms 
which  are  said  to  be  intensively  definable.  This  osten- 
sive  definition  will  be  only  a  special  instance  of  a  form 
of  definition  involving  the  complex  relation  'means  what 
is  indicated  by'— a  relation  which  is  involved  in  any 
attempt  to  define  a  proper  name  by  means  of  a  de- 
scriptive name.    The  particular  force  of  the  notion  of 
ostensive  definition  will  now'  be  explained,  and  it  will 
be  found  to  apply  both  to  an  adjectival  and  to  a  sub- 
stantival name.   The  ordinary  proper  name  applies  to 
an  object  whose  existence  extends  over  some  period  of 
time  and  generally  throughout  some  region  of  space. 
The  appearance  of  such  an  object  in  perception  (or 
rather  of  some  spatially  or  temporally  limited  part  of 
that  object)  provides  the  necessary  condition  for  im- 
posing a  name  in  the  act  of  indicating,  presenting  or 
introducing  the  object  to  which  the  name  is  to  apply, 
and  this  it  is  that  constitutes  ostensive  definition.    In 
extending   the   notion  of   a  proper  name   to   certain 
adjectives  our  justification  is  that  ultimately  a  simple 
adjective-name— such  as  red— cannot  be  defined  ana- 
i  lytically  but  only  ostensively.    Theoretically,  we  must 
suppose  that  any  name,  singular  or  general,  proper  or 
descriptive,   substantival  or  adjectival,   has    originally 
been  imposed  on  a  particular  occasion  by  a  particular 
person  or  group  of  persons.    In  the  case  of  ostensively 
defined  names,  the  occasion  on  which  definition  is  pos- 


sible must  be  one  on  which  the  object  is  actually  pre- 
sented. When,  however,  the  meaning  or  application  of 
such  a  name  has  afterwards  to  be  explained,  or  so-to- 
speak  redefined,  the  only  direct  method  is  to  secure 
for  the  enquirer  another  presentation  of  the  object  in 
question.  Thus  John  Smith,  having  been  presented  to 
his  family  at  birth — which  we  may  take  to  be  the  occa- 
sion on  which  the  name  was  imposed — must  be  pre- 
sented again  to  the  person  ignorant  of  its  application. 
Hence,  in  introducing  a  man  under  the  name  John 
Smith,  we  are  using  the  same  ostensive  method  as  was 
required  in  the  original  definition,  but  such  mention  of 
the  name  does  not,  properly  speaking,  constitute  defi- 
nition. We  are  stating,  in  effect,  the  proposition — which 
is  not  merely  verbal — that  *the  person  introduced  is 
identical  with  the  person  upon  whom  the  name  was 
originally  imposed.'  This  case  of  an  ordinary  substan- 
tival proper  name  is  analogous  to  that  of  an  adjectival 
name — say  cochineal — which  originally  could  only  have 
been  ostensively  defined,  and  which  must  therefore 
be  ostensively  redefined  for  the  person  ignorant  of  its 
application,  in  the  form  'the  colour  of  this  presented 
object  is  identical  with  that  upon  which  the  name  cochi- 
neal was  originally  imposed' — a  statement  which  again 
is  not  merely  verbal.  When  a  proper  name  is  called 
arbitrary,  this  arbitrariness  attaches  only  to  the  original 
act  of  imposition ;  but,  when  the  application  of  the  name 
is  afterwards  explained,  such  explanation  is  no  longer 
arbitrary,  since  to  be  correct  the  real  proposition  that 
the  substantive  or  adjective  presented  is  identical  with 
that  upon  which  the  name  was  originally  imposed,  must 
hold  good,  and  this  statement  may  be  either  true  or 


96 


CHAPTER  VI 


97 


false,  apart  from  linguistic  convention.  Furthermore, 
when  ostensive  definition  is  employed,  it  must  be  ob- 
served that  we  do  not  say  that  the  proper  name  means 
what  is  mean^  by  such  a  phrase  as  'the  object  to  which 
I  am  pointing'  (which  after  all  is  only  an  instance  of  a 
descriptive  phrase),  but  we  say  that  the  proper  name 
means  what  is  indicated  hy  the  descriptive  phrase  'the 
object  to  which  I  am  pointing.'  For  it  is  obvious  in 
this  case,  as  in  the  more  general  account  of  a  descrip- 
tive phrase,  that  however  we  may  further  explicate 
the  meaning  of  the  phrase  'the  object  to  which  I  am 
pointing,'  the  substituted  phrase  would  not  have  the 
nature  of  a  proper  name  but  necessarily  of  a  descrip- 
tive name. 

When,  then,  finally  we  agree  with  the  general  posi- 
tion of  the  best  logicians  that  the  proper  name  (as  Mill 
says)  is  non-connotative,  this  does  not  amount  to  say- 
ing that  the  proper  name  is  non-significant  or  has  no 

[  meaning;  rather  we  find,  negatively,  that  the  proper 
name  does  not  mean  the  same  as  anything  that  could 
be  meant  by  a  descriptive  or  connotative  phrase ;  and 

0.  positively,  that  it  does  precisely  mean  what  could  be 
^W2Va/^^  by  some  appropriate  descriptive  phrase.  This 
exposition  holds  both  for  the  names  of  objects  which 
can  be  presented  and  thus  ostensively  defined ;  and  also 
for  the  names  of  objects  removed  in  time  or  place,  for 
the  definition  of  which  a  descriptive  phrase  (which  is 
other  than  ostensive)  must  be  employed. 


CHAPTER  VII 

GENERAL  NAMES;  DEFINITION  AND  ANALYSIS 

§  I.  Having,  in  the  preceding  chapter,  distinguished 
the  different  kinds  of  articles,  we  now  turn  to  a  common 
characteristic  in  the  use  of  an  article,  namely,  its  attach- 
ment to  a  general  name.  The  general  name  has  usually 
been  differentiated  by  reference  to  number,  and  roughly 
defined  as  a  name  predicable  of  more  than  one  object. 
In  fact,  however,  there  are  general  names  such  as  *  in- 
teger between  3  and  4 '  or  '  snake  in  Ireland '  that  are 
predicable  of  no  object,  while  '  integer  between  3  and 
5  '  and  '  pole-star '  are  general  names  predicable  of  only 
one  object.  There  is  therefore  nothing  in  the  meaning 
of  a  general  name  which  could  determine  the  number 
of  objects  to  which  it  is  applicable.  Rejecting  this  re- 
ference to  number,  we  may  point  out  that  a  universal 
characteristic  of  the  general  name  is  its  connection  with 
the  article — the  use  of  the  grammatical  term  'article' 
being  extended  to  include  this,  that,  some,  every,  any, 
etc.  All  terms  of  this  kind  serve  to  determine  the  in- 
tended application  of  reference  in  a  proposition,  and 
hence  might  more  properly  be  called  applicatives  or  I 
selectives.  Now  a  general  name  is  distinguished  as  that  I 
to  which  any  applicative  can  be  significantly  prefixed: 
thus  the  applicative,  on  the  one  hand,  requires  a  general 
name,  while,  on  the  other  hand,  it  follows  from  the 
essence  of  the  general  name  that  to  it  any  applicative  is 

J.L.  7 


98 


CHAPTER  VII 


significantly  attachable.  And  a  further  and  connected 
characteristic  of  the  general  name  is  that  it  can  always 
be  used  in  the  plural,  or,  in  fact,  with  any  numerical 

prefix. 

If  now  we  further  consider  the  typical  applicatives 
'  every  '  and  '  some,'  we  find  that  in  the  last  analysis 
they  entail  the  conception  of  unlimited  generality,  and 
language  reflects  this  unlimitedreference  by  uniting  these 

applicatives  with  the  substantival  term.  '  thing '  to  form 
the  single  words  '  everything '  and  '  something,'  where 
the  word  'thing'  stands  only  for  abstract  generality 
without  any  such  limitation  as  could  possibly  be  defined 
by  adjectival  characterisation.     If  any  account  of  the 
name  '  thing '  here  could  be  given  in  philosophical  ter- 
minology,  we  should  have  to  say  that  '  thing '  stands 
I  for  the  category  of  all  categories ;  or  more  precisely, 
that  the  generality  implied   in   this  use  of  the  word 
*  thing'  is  so  absolute  that  it  must  include  all  logical 
categories,  such  as  proposition,  adjective,  substantive, 
etc.    On  the  other  hand,  the  general  term  with  which 
ordinary  logic  is  almost  exclusively  concerned,  is  re- 
moved by  two  steps  from  this  absolute  generality:  for 
example,   the  general  term    'man'  comes  under  the 
logical  category  *  substantive,'  which  itself  is  a  limitation 
of  the  absolute  generality  peculiar  to  the  word  'thing.' 
Thus  in  philosophical  logic  we  ought  to  be  prepared 
to  define  the  category  '  substantive '  by  some  limiting 
characterisation  within  the  absolute  general  *  thing'; 
and  similarly  of  the  category  'adjective'  or  'proposition,' 
etc.     And  again,  in  ordinary  logic,  any  general  name 
is  understood  to  be  defined  first,  as  coming  under  an 
understood  category— in  particular  the  category  sub- 


GENERAL  NAMES;  DEFINITION  AND  ANALYSIS    99 

stantive — and  secondly  as  delimited  by  a  certain  adjec- 
tival characterisation.  As  regards  any  specified  general 
substantive-name,  its  nature  as  general  is  brought  out 
by  showing  its  meaning  to  be  resolvable  into  some 
specific  adjective  or  conjunctionof  adjectives,  juxtaposed 
to  the  category  'substantive'  itself. 

The  consideration  that  to  the  general  name  any 
applicative  can  be  prefixed,  distinguishes  it  from  the     r 
singular  name,  whether  descriptive  or  proper.   We  have    jj-. 
now  tobring  out  a  characteristic  shared  incommon  by  the 
general  name  and  the  singular  descriptive  name,  which 
distinguishes  them  both  from  the  proper  name :  namely 
that  the  two  former  always  (except  in  the  case  of  the    - 
absolute  general  name  '  thing')  contain  in  their  meaning 
an  adjectival  factor,  whereas  the  strictly  proper  name   A 
contains    in  its    meaning  no   adjectival    factor.     This 
account  may  seem  a  somewhat  arbitrary  way  of  settling 
the  prolonged  controversy  initiated  by  Mill  as  to  whether    ]f 
proper   names  are   non-connotative.     In   default  of  a 
definition  of  a  proper  name,  however,  it  is  impossible 
to  decide  whether  any  given  name,  such  as  London,  is 
to  be  called  proper.    I  propose,  therefore,  to  define  the 
word  proper  as  equivalent  to  non-connotative,  non-de-     i<^ 
scriptive  or  non-significant  (since  these  three  terms  are  , 
themselves  synonymous),  and  the  only  debatable  point  ' 
which  remains  is  as  to  whether  any  names  can  properly 
be  called  'proper\'    A  similar  remark  applies  to  the 
question  whether  all  general   names  are  connotative, 
since  it  would  seem  necessary  to  define  a  general  name 

^  This  brief  account  of  the  Proper  Name  has  been  discussed  and 
qualified  in  the  preceding  chapter;  but  for  the  present  purpose  it  is 
adequate. 

7—2 


9v 


100 


CHAPTER  VII 


i'-n 


(except  the  absolutely  general  name  '  thing*)  as  equi- 
valent to  one  which  is  connotative. 

§  2.  At  this  point  we  must  enquire  into  the  precise 
^v^^iiiurfv :  meaning  of  the  word  connotation^  and  this  enquiry- 
necessitates  the  introduction  of  the  more  general  notions 
of  intension  and  extension.  While  extension  stands  for 
a  set  of  substantives,  intension  stands  for  a  set  of  adjec- 
tives ;  and  moreover  the  two  terms  are  used  in  correlation 
with  one  another,  the  substantives  comprised  in  the 
extension  being  characterised  by  the  adjectives  com- 
prised in  the  intension.  Now  of  all  the  adjectives  that 
may  characterise  each  of  a  set  of  substantives,  a  certain 
sub-set  will  have  been  used  for  determining  the  appli- 
cation of  the  general  name,  and  the  range  of  extension 
thereby  determined  constitutes  the  denotation  of  that 
J  name.  Thus  the  specific  function  of  connotation  is  that 
'it  is  used  to  determine  denotation;  and  hence  any  other 
adjectives  that  may  characterise  all  the  substantives 
comprised  in  the  denotation,  do  not  determine  the 
denotation,  but  rather  are  determined  by  it.  The  entire 
and  often  innumerable  conjunction  of  adjectives  deter- 
mined by  the  denotation  has  been  called  by  Dr  Keynes, 
I  the  Comprehension.  Thus  Dr  Keynes's  exposition  may 
I  be  summed  up  in  the  statement  that  connotation  in  the 
first  instance  determines  denotation,  which  in  its  turn 
determines  comprehension.  The  controversy  on  the 
question  of  what  adjectives  should  be  included  in  the 
connotation  of  the  general  term,  has  arisen  from  the  false 
supposition  that  the  logician  starts  with  a  known  range 
of  denotation,  and  that  with  this  datum  he  has  to  dis- 
cover, amongst  the  known  common  characters  peculiar  to 
the  members  of  the  class,  those  which  shall  constitute  the 


GENERAL  NAMES;  DEFINITION  AND  ANALYSIS    loi 

connotation.  But  this  is  reversing  the  order  of  priority, 
since  there  is  no  means  of  delimiting  the  range  of  de- 
notation of  a  term  otherwise  than  by  laying  down  a  finite 
enumeration  of  adjectives  which,  taken  together,  con- 
stitute the  test  for  applying  the  name.  This  test,  which 
is  provided  by  the  connotation,  may  have  remained 
unchanged,  or  have  varied  in  the  course  of  time,  or 
again  have  been  technically  determined  by  science  in 
such  a  way  as  to  conflict  more  or  less  with  common 
usage ;  but  in  every  case  what  constitutes  the  connotation 
is  invariably  a  conjunction  of  adjectives  which,  within 
these  limits  of  time  or  context,  is  used  to  determine: 
the  application  of  the  name.  ' 

We  are  indebted  to  Mill  for  an  elaborate  and  com- 
plete treatment  of  connotative  names,  including  the 
connection  between  connotation  and  definition ;  but  the 
formula  that  the  definition  of  a  general  name  is  the  un- 
folding of  its  connotation,  must  be  corrected  by  the 
added  reference  to  the  substantival  element,  since  con- 
notation by  itself  is  purely  adjectival.  It  is  obviously 
impermissible,  for  instance,  to  substitute  for  the  sub- 
stantival name  *  a  man '  a  mere  adjective  *  human '  or 
the  abstract  term  '  humanity  ' ;  rather  '  a  man  '  should 
be  defined  as  *a  human  being'  or  *  a  being  characterised 
by  the  attribute  humanity.'  In  other  words,  a  substan-  i 
tive-name  must  be  so  defined  as  to  show  that  it  is  sub- 
stantival. The  word  *  being '  here  has  the  force  of  the 
word  *  substantive,'  so  that  pushed  to  its  logical  con- 
clusion, *  a  man  '  should  be  defined  as  *  a  human  sub- 
stantive ' ;  and  it  is  the  adjective  *  human '  which  alone 
may  require  further  analysis  in  our  definition,  not  the 
word  *  being'  or  *  substantive.' 


102 


CHAPTER  VII 


The  same  principle  will  apply  to  the  definition  of  a 
general  adjectival  name,  if  it  is  admitted  that  certain 
adjectives  and  relations  (which  may  be  called  secondary) 
can  be  properly  predicated  of  other  adjectives  (which  in 
this  connection  might  be  called  primary).  For  example, 
in  the  proposition :  *  the  unpunctuality  of  his  arrival 
was  annoying,'  we  appear  to  be  predicating  of  the 
primary  quality  or  adjective  represented  by  the  abstract 
name  'unpunctuality'  the  further  or  secondary  adjective 
*  annoying.'  By  a  secondary  adjective  we  mean  an 
adjective  of  an  adjective,  and  if  the  primary  adjective  is 
expressed  grammatically  as  a  substantive,  the  secondary 
adjective  would  be  expressed  grammatically  as  an  ad- 
jective, but  if  the  primary  adjective  retains  its  gram- 
matical form  as  an  adjective,  then  its  secondary  adjective 
is  expressed  by  an  adverb,  which  is  logically  equivalent 
to  an  adjective  of  an  adjective.  Thus  the  following 
series  of  propositions :  *A  is  moving,'  *the  movement 
of  A  is  rapid,'  '  the  rapidity  of  the  movement  of  A  is 
surprising' — involve  the  primary  adjective  'moving,' 
the  secondary  adjective  'rapid,'  and  what  we  must  here 
call  the  tertiary  adjective  '  surprising.'  When  the  primary 
adjective  '  moving '  retains  its  adjectival  form,  the 
secondary  adjective  predicated  of  it  would  be  expressed 
as  an  adverb;  thus  'A  is  moving  rapidly';  or  when 
the  secondary  adjective  '  rapid '  retains  its  adjectival 
form,  the  tertiary  adjective  predicated  of  it  is  again 
expressed  as  an  adverb,  as  in  'A's  movement  was  sur- 
prisingly rapid.'  These  examples  seem  to  confirm  the 
view  that  adjectives  can  properly  be  predicated  of  ad- 
jectives as  such.  If  this  analysis  is  correct,  we  should 
expect  to  find  certain  general  adjectival  names  (ana- 


GENERAL  NAMES ;  DEFINITION  AND  ANALYSIS    103 

logous  to  general  substantival  names)  whose  meaning 
could  be  elicited  in  terms  of  the  secondary  adjectives 
implied  in  their  use.  Thus  to  take  Mill's  example,  the 
name  'fault'  which  is  predicable  of  such  primary  qualities 
or  adjectives  as  'laziness,'  'unpunctuality,'  'untidiness,' 
is  predicable  of  such  qualities  of  conduct  on  the  ground 
that  these  are  characterised  by  the  further  or  secondary 
quality  '  faultiness.'  In  this  way  the  definition  of  a 
general  adjectival  name  would  be  formulated  in  terms 
of  the  secondary  adjective  or  conjunction  of  adjectives, 
constituting  its  connotation,  juxtaposed  to  the  category 

adjective  itself. 

§  3.    Turning  now  to  another  aspect  of  our  topic, 
we  shall  consider  the  nature  of  definition  only  in  the 
form  of  bi-verbal  substitution,  where  the  reference  is 
restricted  to  words  or  phrases;  in  contradistinction  to 
ideas  or  things,  which  some  philosophers  have  under- 
taken  to  define.    The  question    of  the   definition  of 
words  requires  a  wider  treatment  than  that  generally 
accorded  to  it  in  logic ;  for  under  the  influence,  I  pre- 
sume, of  scholastic  doctrine,  definition  has  been  tacitly 
restricted  to  the  case  of  substantive  terms,  to  which  the 
traditional  formula  '  per  differentiam  et  genus '  is  alone 
applicable.    It  seems,  indeed,  as  if  logicians  had  shrunk 
in  terror  from  the  task  of  defining  other  than  substantive- 
terms,  setting  aside  preposition  words,  conjunction  words, 
pronominal  words  and  even  adjectives,  as  if  these,  as 
such,  were  outside  the  scope  of  logical  definition.    The 
problem  of  definition,  it  is  clear,  must  extend  to  2«Z  word, 
however  it  may  be  classified  by  grammar.    We  must 
certainly  come  to  some  mutual  understanding  of  the 
meanings  of  such  words  as  '  and,'  '  or,'  '  if,'  inasmuch  as 


OtfiWi^***: 


104 


CHAPTER  VII 


these  and  many  other  such  words  have  a  meaning,  the 
understanding  of  which  is  essential  in  the  pursuit  of 
logic.  It  is  frequently  impossible,  however,  to  define  a 
word  taken  in  isolation,  and  in  such  cases,  we  must 
construct,  as  that  which  is  to  be  directly  defined,  a  verbal 
phrase  containing  the  word.  We  should  not  perhaps  go 
far  wrong  if  in  every  such  case  we  took  the  completed 
propositional  phrase  as  that  which  is  to  be  defined, 
although  it  is  often  possible  of  course  to  take  only  some 
part  of  a  proposition  and  succeed  in  meeting  the  re- 
j  quirements.  We  repeat  then  that  our  problem  is  how 
j  to  define  a  given  verbal  phrase ;  and  the  answer  is  to 
substitute  for  it  another  verbal  phrase.  This  is  the 
complete  and  quite  universal  account  of  the  procedure 
of  definition,  which  justifies  our  restriction  of  the  topic 
to  bi-verbal  definition ;  its  obvious  purpose  is  fulfilled 
if  the  substituted  phrase  is  understood.  (Cp.  preceding 
chapter.) 

In  this  connection  it  is  worth  noting  that,  when 
what  has  to  be  defined  is  a  verbal  phrase  rather  than 
a  single  word,  we  may  italicise — so  to  speak — that  part 
of  the  phrase  for  which  an  explanation  is  asked.  In 
such  cases  the  remaining  components  of  the  phrase 
may  be,  and  generally  ought  to  be,  repeated  in  the 
phrase  constituting  the  definition.  In  symbols,  let  us 
say  that  the  phrase  'abed'  requires  definition,  where 
the  components  'be'  are  combined  in  the  whole  'abed'  in 
such  way  that  the  combination  'abed'  is  not  understood. 
Suppose  the  symbol  '  apqd'  to  represent  our  definition ; 
then  we  shall  have  defined  'abed'  (where  the  component 
'be'  required  explanation),  by  the  phrase  ' apqd'  in 
which  'pq'  explicitly  replaces  'bc^  and  is  offered  in  expla- 


GENERAL  NAMES;  DEFINITION  AND  ANALYSIS    105 

nation.    A  definition  such  as  that  symbolised  above  is 
rejected   in  current  logical  text-books  on  the  ground 
that  it  commits  the   sin  of  tautology ;  for  it  repeats 
verbatim  the  symbol   'ad'  in  the  definition  given  as 
explanation  of  '  abed.'     But  this  mode  of  definition,  so   ^ 
■j     far  from    being  a   ground   of   condemnation,   exactly 
answers  in  the  most  adequate  sense  the  requirements. 
The    more   exacdy   we    repeat    in  our   definition  the 
actual  words  and  their  form  of  combination,  used  in  the 
phrase  to  be  explained,  the  more  precisely  do  we  meet 
the  demands  for  an  explanation.     Typical  instances  of 
this  principle  will  occur  to  anyone  who  reflects  on  the 
subject.    It  follows  that  no  general  or  merely  formal 
criticism  of  a  definition  can  be  made  by  any  logical  rule ; 
the  question  whether  any  proposed  definition  is  good 
or  not  being  entirely  relative  to  the  enquirer's  knowledge 
and  ignorance  of  meanings.  This  by  no  means  precludes 
the  possibility  of  definitions  which  would  be  generally 
useful,  because  any  obscurity  or  ambiguity  which  one 
person  might  feel  is  likely  to  be  felt  by  others ;  a  litde 
common  sense  is  in  general  all  that  is  necessary. 

This  account  leads  at  once  to  one  conclusion,  which 
is  perhaps  tacitly  understood  by  all  logicians  and  philo- 
sophers; i.e.,  that,  inasmuch  as  the  only  way  to  explain 
one  verbal  phrase  is  to  substitute  another,  therefore  no 
successive  chain  of  explanatory  phrases  can  serve  the 
purpose  of  ultimate  explanation,  if  that  chain  is  endless. 
Hence  perhaps  the  important  point  in  the  theory  of' 
explanatory  definition  is  that  it  must  stop.    In  other  i 
words,  by  a  shorter  or  longer  process,  every  definition 
must  end  with  the  indefinable.    A  certain  misunder- « 
standing  as  to  what  in  logic  is  meant  by  the  indefinable 


^A-_ 


io6 


CHAPTER  VII 


must  here  be  removed ;  for  it  has  been  frequently  sup- 
posed that  the  indefinable  means  that  which  is  admittedly 
not  understood.  But  so  far  from  meaning  the  *  not- 
understood,'  the  indefinable  means  that  which  is  under- 
stood ;  and  philosophy  or  logic  may  ultimately  adopt  a 
term  as  indefinable  only  where,  because  it  is  understood, 
it  does  not  require  a  further  process  of  definition.  Phi- 
losophy must  never  stop  with  the  indefinable,  in  the 
sense  of  reaching  a  component  of  thought  expressed 
obscurely  or  with  an  admitted  margin  of  doubt  as  to 
meaning.  The  indefinable  does  not  therefore  mean  that 
which  is  presented  as  having  no  understood  meaning, 
but  that  whose  meaning  is  so  directly  and  universally 
understood,  that  it  would  be  mere  intellectual  dishonesty 
to  ask  for  further  definition. 

§  4.  There  has  been  practical  unanimity  in  re- 
garding definition  as  a  process  which  essentially  in- 
volves analysis.  We  have  above  reached  the  idea  of 
an  indefinable;  and  it  has  been  almost  universal,  I 
believe,  to  regard  the  indefinable  as  equivalent  to  what 
is  incapable  of  analysis — a  view  which  is  obvious  if 
definition  and  analysis  coincide.  But,  if  definition  is 
not  to  be  merely  equated  to  analysis,  then  we  may  pause 
before  we  regard  anything  as  being  indefinable  on  the 
ground  of  its  being,  in  some  proper  sense,  unanalysable. 
In  my  own  view,  definition  assumes  so  many  varied 
forms  that  its  equivalence  to  analysis  seems  to  be  highly 
dubious. 

Before  entering  into  further  detail  on  this  point  it  will 
be  well  to  consider  what  is  meant  by  the  word  analysis 
in  philosophy  and  logic.  Associated  with  it  we  often 
find  a  reference  to  parts  and  wholes:  thus,  analysis  is 


GENERAL  NAMES;  DEFINITION  AND  ANALYSIS    107 

often  said  to  mean  the  separation  of  a  whole  into  its 
I  parts.     But   '  separation '   does  not  adequately  repre- 
f  sent  the  process.    For  instance,  grammatical  analysis 
does  not  mean  taking  the  several  words  of  a  sentence 
and  determining  the  part  of  speech  and  inflection,  etc., 
of  each  word,  but  its  object  is  to  show  how  the  signifi- 
cance of  the  sentence  is  determined  by  the  mode  in 
which  the  several  words  are  combined.     Similarly  the 
analysis  of  a  psychosis  is  not  merely  the  cataloguing  of 
such  elements  as  knowing,  feeling,  acting,  but  rather 
the  representation  of  the  essential  nature  of  a  given 
psychosis  as  determined  by  the  mode  in  which  these 
factors  combine.    What  holds  of  grammatical  and  psy- 
chological analysis  holds  of  every  kind  of  analysis ;  and, 
since  the  important  process  is— not  the  mere  revelation 
of  the  parts  contained— but  rather  the  indication  of  their 
mode  of  combination  within  the  whole,  analysis  is  better 
defined  as  the  exhibition  of  a  given  object  in  the  form 
of  a  synthesis  of  parts  into  a  whole.    In  this  way  we  can 
say  that  any  process  of  analysis  can  also  be  described 
as  a  process  of  synthesis ;  but  this  does  not  amount  to 
saying  that  analysis  means  the  same  as  synthesis,  any 
more  than  that  the  relation  'grandfather'  is  the  same 
as  the  relation  '  grandson,'  although  the  fact  that  A  is 
the  grandfather  of  B  is  the  same  fact  as  that  B  is  the 
grandson  of  A.    In  short,  analysis  is  the  inverse  of  syn- 1 
thesis ;  i.e.  when  the  whole  X  is  analysed  into  its  several  I 
components  a,  b,  c,  d\  then  a,  b,  c,  d  have  to  X  the  in- 
verse relation  which  X  has  to  a,  b,  c  and  d.     In  this 
way  it  is  clear  that  to  analyse  X  simply  means  the  same 
as  to  exhibit  X  as  a  synthesis. 

Now  instead  of  taking  X  as  a  term  to  be  defined. 


io8 


CHAPTER  VII 


and  exhibiting  it  as  a  synthesis  of  a,  b,  c  and  d^  let 
us  take  the  term  a  and  define  it  by  showing  how  it 
functions  in  a  whole  X  where  it  is  combined  with  b,  c 
and  d.     This  points  to  two  modes  of  definition,  viz. 
analytic  and  synthetic.     In  analytic  definition  we  pass 
d    from  an  unanalysed — i.e.  an  apparently  simple — to  an 
analysed  equivalent;  in  synthetic  definition  we  exhibit 
the  nature  of  what  is  simple,  not  by  representing  it  as 
a  complex,  but  by  bringing  it  into  synthetic  connec- 
tion in  a  complex  of  which  it  is  a  component.    Or  more 
shortly:  analytic  definition  is  explaining  a  complex  in 
terms  of  its  components,  and  synthetic   definition   is 
explaining  components  in  terms  of  a  complex.    A  few 
illustrations  will  help  to  make  clear  exactly  what  we 
mean  by  synthetic  definition.    Take,  in  arithmetic,  the 
definition  of  'factor'  or  'multiple.'    We  first  construct 
a  certain  complex,  involving  integers  illustratively  sym- 
bolised by  Uy  by  c] — namely,  the  proposition  axb  =  c\ 
this  complex,  being  understood^  is  used  to  define  the 
terms  that  require  definition,  and  the  definition  assumes 
the  following  form:  a  is  said  to  be  a  factor  of  Cy  or 
c  is  said  to  be  a  multiple  of  ^,  when  the  relation  ex- 
pressed in  the  proposition  ay.b  =  c  holds.    Here  we  do 
notresolvethe  meaningsof  the  terms 'factor 'or 'multiple' 
into  their  simple  components  of  meaning,  but  define 
them  by  showing  in  what  way  they  enter  as  components 
into  the  understood  construct — viz.  the  particular  equa- 
tional  proposition.  Again  taking  the  algebraic  definition 
of  '  logarithm ' :    we   begin  by  constructing  a  certain 
whole — expressed  for  convenience  in  terms  of  general 
illustrative  symbols  ^,  /,  /, — viz.  the  proposition  that 
'  b  to  the  power  /  equals  p. '    This  constructed  complex — 


GENERAL  NAMES;  DEFINITION  AND  ANALYSIS    109 

the  nature  of  which  is  presumed  to  be  understood — is 
the  medium  in  terms  of  which  the  notion  of  a  logarithm 
will  be  defined  as  follows:  '/  is  the  logarithm  of/  to  the 
base  b '  means  what  is  meant  by  the  proposition :  '  ^  to 
the  power  /  equals  /.'     Again  take  the  definition  of 
'sine':  here,  starting  with  an  angle  A  bounded  at  the 
point  O  by  the  lines  OXy  ORy  we  make  the  following 
construction ;  taking  some  point  P  in  ORy  and  dropping 
the  perpendicular  PM  upon  OXy  then  the  sine  of  A 
means  the  ratio  of  MP  to  OP,    The  constructed  com- 
plex here,  namely  of  a  certain  right-angled  triangle,  has 
first  to  be  indicated  and  understood,  and  by  its  means 
we  are  enabled  to  define  what  may  be  called  a  component 
of  this  complex.    Another  example  of  this  form  of  de- 
finition is  afforded  in  any  attempt  to  define  the  words 
'substantive'  and  'adjective.'    Here  we  may  presuppose 
that  the  notion  of  '  proposition  '  is  understood — e.g.  as 
that  of  which  'true'  or  'false'  may  be  significantly  pre- 
dicated— and  further  presupposing  that  the  notion  of 
characterisation  is  understood,  the  first  account  of  sub- 
stantive and  adjective  will  be  that  they  are  combined 
in  a  proposition  in  a  mode  expressible  either  in  the 
form  that  *  a  certain  substantive  is  characterised  by  a 
certain  adjective, 'or  that 'a  certain  adjective  characterises 
a  certain  substantive.'     Here  we  give  at  the  same  time 
the  definitions  of  substantive  and  of  adjective  by  showing 
how,  as  components  in  the  whole,  i.e.  the  proposition, 
they  have  to  be  combined\ 

§  5.    The  general  notions  of  analysis  and  synthesis 

^  All  the  definitions  occurring  in  a  symbolic  system,  whether 
Logical  or  Mathematical,  should  in  my  view  be  synthetic  (in  the 
above  sense)  and  never  analytic. 


no 


CHAPTER  VII 


are  often  explained  in  terms  of  parts  and  whole ;  but 
these  latter  terms  should  be  used  for  a  process  essen- 
tially different  from  analysis.  At  least  three  processes 
which  are  commonly  confused  are  here  to  be  carefully 
distinguished,  viz.,  partition,  resolution,  and  analysis 
proper;  probably  other  more  subtle  variations  must 
finally  be  recognised. 

(I)  By  partition  is  meant  transforming  what  is  first 

presented  as  a  mere  unit  by  exhibiting  it  in  the  form  of 
a  whole  consisting  of  parts ;  it  is  perhaps  more  generally 
defined  as  the  process  of  dividing  a  whole  into  its  part ; 
but  it  is  of  the  first  importance  to  point  out  that  until  a 
thing  is  presented  as  having  parts,  it  cannot  be  said  to 
be  a  whole.  This  conception  of  part  and  whole  should 
be  stricdy  limited  to  three  types  of  cases :  ( i )  to  an  ag- 
gregate, and  toa  number  as  the  adjective  of  an  aggregate ; 
(2)  to  what  occupies  space  and  to  the  space  which  it 
occupies ;  (3)  to  what  fills  time  and  to  the  time  filled. 
These  three  cases  bring  out  the  essential  nature  of  the 
conception,  viz.,  that  the  parts  must  always  be  conceived 
as  homogeneous  with  one  another  and  with  the  whole 
which  they  constitute;  and  further  that  a  certain  character 
called  magnitude  is  predicable  of  any  whole,  the  measure 
of  which  is  equal  to  the  sum  of  the  magnitudes  predicable 
of  the  parts'. 

(ii)  Next,  consider  the  term  resolution.  This  is  very 
shortly  explained  as  the  process  of  exhibiting  a  com- 
posite in  terms  of  its  components ;  but  such  a  definition 
is  open  to  the  same  kind  of  criticism  as  we  have  levelled 

^  No  intensive  or  qualitative  characteristic  of  an  object  can  be 
regarded  as  a  'whole'  of  which  a  magnitude  can  be  predicated  by 
addition  of  the  magnitudes  of  its  'parts.' 


GENERAL  NAMES;  DEFINITION  AND  ANALYSIS    in 

against  the  popular  definition  of  partition,  and  by  a 
parallel  emendation  we  shall  say  that  resolution  means  ; 
the  exhibition  of  what  is  presented  as  simple  in  the  form  ; 
of  a  composite  of  which  the  components  are  assigned.  1 
An  example  of  psychological  interest  is  the  resolution 
of  a  chord  heard  into  its  component  notes,  or  again  of 
a  note  into  its  component  tones,  where  in  either  case 
the  combination  describes  or  accounts  for  the  sound  as 
heard.    While,  on  Helmholtz's  theory  of  auditory  sensa- 
tions, the  physiological  process  here  involved  would  be 
represented  as  a  whole  capable  of  partition,  it  remains 
none  the  less  true  that  psychological  apprehension  pre- 
sents the  sound  as  a  composite  to  be  resolved. 

Then  thirdly,  I  should  restrict  the  word  analysis  to  (uO 
a  process  which  is  distinctively  logical,  and  which  as- 
sumes its  simplest  form  when  we  combine  various 
adjectives  as  predicable  of  one  and  the  same  substantive, 
by  means  of  the  mere  conjunction  *and.'  A  simple  ex- 
ample will  bring  out  the  distinction  between  resolution 
and  analysis.  We  have  shown  what  is  meant  by  resolving 
a  note  into  its  component  tones.  Now  the  character  of 
the  note  is  described  under  certain  aspects,  such  as 
pitch,  intensity  and  timbre,  and  this  constitutes  an 
analysis  of  its  character.  These  three  characteristics 
are  predicated  of  the  sound,  not  in  the  sense  of  re- 
solving the  sound  into  various  component  sounds,  but 
in  the  sense  of  characterising  the  sound  itself,  whether 
it  be  composite  or  simple.  Thus  taking  timbre  for 
instance  as  one  of  the  constituent  characters,  if  a  note 
contains  three  partial  tones  this  would  count  as  3  in  its 
resolution  but  i  only  in  its  analysis  ;  if  on  the  other 
hand  the  note  were  simple — i.e.  contained  only  one 


f 


112 


CHAPTER  VII 


"3 


component  tone — this  would  count  as  i  in  the  resolution, 
while  the  analysis  of  this  single  note  yields  the  same  3 
characters  (pitch,  timbre  and  intensity)  as  that  of  the 
composite  note.  The  value  of  this  illustration  is  that  it 
conclusively  disposes  of  the  assumption  that  a  plurality 
of  predications  characterising  an  object  depends  at  all 
upon  its  partition  or  resolution;  that  is,  upon  regarding 
the  object  either  as  a  whole  consisting  of  parts  or  as 
a  composite  resolvable  into  components. 

So  far  we  have  considered  that  form  of  analysis 
which  exhibits  its  object  as  a  synthesis  of  constituents 
conjoined  by  the  conjunction  'and,'  and  yielding  what 
will  be  termed  a  compound.  In  contrast  to  a  compound 
synthesis  we  must  consider  also  what  must  be  called  a 
complex  synthesis — namely  one  in  which  the  material 
constituents  are  heterogeneous,  including  substantives 
of  different  kinds  and  adjectives  of  different  order- 
monadic,  diadic,  triadic'— which,  qua  heterogeneous, 
are  united  in  different  modes  from  that  of  simple  con- 
junction. Thus  the  word 'courageous'  yields  the  complex 
synthesis  *not  flinching  from  danger' ;  where  the  material 
constituents  are  'danger'  and  'flinching  from,'  of  which 
the  former  is  expressed  substantivally  and  the  latter  as 
a  diadic  adjective.  From  this  fairly  simple  example  it 
will  be  seen  that  the  possible  forms  of  complexity  that 
analysis  may  yield  are  inexhaustible.  The  discussion  of 
this  topic  will  be  continued  from  a  somewhat  different 
aspect  in  a  subsequent  chapter  on  relations. 

1  See  Chapter  IX. 


CHAPTER  VIII 

ENUMERATIONS  AND  CLASSES 

§  I.    An  enumeration  is  an  assignment  of  certain 
items  which  may  be  said  to  be  comprised  in  the  enumera- 
tion.   We  attach  therefore  to  an  enumeration  the  con- 
ception of  unity  as  applied  to  the  whole  along  with 
plurality  of  the  items  comprised  in  this  whole.   In  naming 
the  items  to  be  comprised  in  an  enumeration  as  a,  b,  c, 
d,  e,  for  instance,  it  will  generally  be  implied  that  we 
shall  not  repeat  any  item  previously  named;  also,  that 
the  order  of  assignment  is  indifferent.    For  the  purposes 
of  elementary  illustration,  we  shall  consider  that  all  the 
items  ultimately  to  be  enumerated  have  a  finitely  as- 
signed number  (say)  1 2 :  which  may  be  named  respec- 
tively ay  b,  c,  d,  eyf.gy  h,  k,  /,  m,  n.    Such  an  enumeration 
might  be  called  our  enumerative  universe.    Thus  taking 
any  'assigned  enumeration  included  in  this  universe,  we 
may  speak  of  the  remainder  to  this  enumeration — by 
which  will  be  meant  the  items  comprised  in  the  universe, 
but  not  comprised  in  the  first  assigned  enumeration. 
The  notion  of  remainder  is  therefore  associated  with 
the  notion  of  not\  although  the  two  must  be  strictly 
distinguished.   The  remainder  to  an  assigned  enumera- 
tion is  the  simplest  function  of  a  single  enumeration 
with  which  we  shall  be  concerned.    We  next  consider 
the  typical  functions  of  two  enumerations — E,  F  say : 


/.M^ 


;*...*.„^«-*^' 


i 


J.  L. 


8 


114 


CHAPTER  VIII 


ENUMERATIONS  AND  CLASSES 


115 


namely  '  E  into  F;  and  '  E  with  F!    By  the  former  is 
meant  the  largest  enumeration  which  is  included  both 
in  E  and  F ;  by  the  latter,  the  smallest  enumeration 
that  includes  both  E  and  F,    Thus,  let  E  be  \_a,  b,  c,  d, 
e,f]  and  let  F  be  \_d,  ej,g,  h~\\  then  'E  into  F'  will  be 
\d,  ej\,  and  'E  with  F'  will  be  \a,  b,  c,  d,  ej,g,  h\ 
Anticipating  elementary  arithmetical  notions,  we  may 
\    at  once  assert  the  generalisations:  first,  that  the  number 
for  '^  into  F'  cannot  be  greater  than  the  number  for 
J  ^  or  the  number  for  F\  and  secondly,  that  the  number 
for  '^5'  with  F'  cannot  be  less  than  the  number  for  E 
or  the  number  for  F.    U  E  and  F  are  identical,  then  in 
every  sense  of  the  word  equals,  E  into  F=E  with  F 
=  £  =  F.     This  represents  one  limiting  case.     If,  on 
the  other  hand,  some  items  comprised  in  E  are  com- 
prised in  F,  the  number  for  'E  with  F'  is  less  than  the 
sum  of  the  numbers  for  E  and  F  respectively,  and  if 
there  are  no  items  comprised  both  in  E  and  in  F,  then 
the  number  for  '  E  with  F'  will  equal  the  sum  of  the 
numbers  for  E  and  F  respectively,  while  the  number 
.for  '^  into  F'  is  zero.     In  the  former  case  E  and  F 
would  be  said  to  be  no^  exclusive,  and,  in  the  latter,  ex- 
I  elusive  of  one  another.    For  the  general  case : 
The  number  for  {E  into/^)  -f  the  number  for  {E  with  F) 
=  the  number  for  ^H-the  number  for  F, 
For  the  purpose  of  further  development,  we  will 
*  abbreviate  the  term  remainder  into  R\    The  symbol  R\ 
i.e.,  remainder  tOy  and  the  prepositions  with,  into  may 
be  called  '  operators,'  because  each  indicates  a  certain 
operation   to  be   performed   upon    one   or   upon   two 
enumerations  by  means  of  which  another  related  single 
enumeration  of  the  same  order  is  to  be  constructed. 


The  relation  of  the  given  operation  to  the  enumeration 
which  is  to  be  constructed  will  be  called  the  relation  of 
yielding.  Now,  noting  that  R  \a,  b,  c,  d,  e,  /]  yields 
[g,  k,  k,  /,  m,  n]  and  that  7?'  [d,  e,  /,  g,  A]  yields  [a,  b,  c, 
k,  /,  m,  n\  we  may  illustrate  the  import  of  these 
operators  in  the  eight  following  formulations  where  the 
symbol  =  stands  {or yields: 

(i)        [a,  b,  c,  d,  e,f]  into       [d,  ej,  g,  h\^\d,  e,f\ 

(2)  R  [rt,  b,  c,  d,  ej\  into  R  [d,  ej,  g,  K\^\_k,  /,  ;«,  «] 

(3)  R  \a,  b,  c,  d,  ej\  into       \d,  ej,  g,  h]  =  lg,  h] 

(4)  [«,  b,  c,  d,  ej\  into  R  \d,  ej,  g,  h\  =  \a,  b,  c] 

where  (i)  with  (2)  with  (3)  with  (4)  =  the  enumerative  universe. 

(5)  R  [a,  b,  c,  d,  e,/]  with  R  [d,  ej,g,  /i]  =  [a,  b,  c,  h,  k,  /,  w,  n\ 

(6)  [«,  b,  c,  d,  e, /]  with      \d,  e, /,g/i]=^ [a,  b,  c,  d,  e, /  g,  A] 

(7)  [a,  b,  c,  d,  e,/]  with  R  [d,  ej,g,  A]=[^,  b,  c,  d,  e,f,  k,  /,  w,  n\ 

(8)  R  [«,  b,  c,  d,  ej\  with       \d,  ej,g,  /i]^[d,  e,  /,  g,  h,  k,  /,  m,  n] 
where  (5)  into  (6)  into  (7)  into  (8)  =  the  enumerative  zero. 

As  regards  these  eight  formulae  we  observe  that 
each  of  the  pairs  (i)  and  (5),  (2)  and  (6),  (3)  and  (7), 
and  (4)  and  (8)  give  two  enumerations  related  the 
one  to  the  other  as  remainder.  What  holds  in  one 
illustration  can  be  formulated  in  general  terms.  Let 
E  and  F  be  any  two  enumerations,  then :  the  operation 
'E  into  F'  yields-the-enumeration-yielded-by  the  opera-  j 
tion  7?'  {RE  with  RF).  Since  the  relation  'remainder  i 
to'  as  also  the  relation  *  yields-what-is-yielded-by '  are  re- 
versible or  symmetrical,  this  single  formula  includes  all 
the  eight  formulae  which  have  been  illustrated  above. 
But,  we  may  for  the  sake  of  emphasis,  express  the 
principle  again  in  eight  formulations  where  =  will  now 
stand  for  *  yields-what-is-yielded-by.' 

S—2 


ii6 


CHAPTER  VIII 


ENUMERATIONS  AND  CLASSES 


117 


(i)  ^  into  F        =the  remainder  to  RE  with  RF 

(2)  RE  into  RF=  „  „  „   E  with  F 

(3)  RE'mtoF     =  „  „  ,,   E  with  RF 

(4)  E'mtoRF     =  „  „  „   ^'^  with  7^ 
where  (i)  with  (2)  with  (3)  with  (4)  =  the  enumerative  universe. 

(5)  RE  with  RF=^  „  „  „   E'mtoF 

(6)  ^withF       =  „  „  ,,   RE  into  RF 

(7)  E  with  RF    =  „  „  „   RE  into  F 

(8)  RE  with  F   =  „  „  ,,   E'mtoRF 
where  (5)  into  (6)  into  (7)  into  (8)  =  the  enumerative  zero. 

From  any  one  of  the  above  eight  formulae  we  can 
read  off  any  other.  Where  any  one  mode  of  constructing 
an  enumeration  is  equivalent  in  the  above  sense  of  = 
to  a  certain  other  mode  of  constructing  an  enumeration, 
it  is  obvious  that  the  equivalence  will  imply  equality  of 
number,  although  the  reverse  does  not  hold  :  that  is, 
we  may  have  equality  of  number  for  two  enumerations 
while  the  items  comprised  in  them  are  not  necessarily 

the  same. 

§  2.  We  have  now  to  consider  how  a  single  enumera- 
tion may  be  taken  as  an  item  to  be  enumerated  along 
with  other  enumerations  so  as  to  constitute  an  enumera- 
tion of  enumerations,  that  is  an  enumeration  comprising, 
as  its  items,  units  which  are  themselves  enumerations. 
This  conception  of  an  enumeration  comprising  enumera- 
tions must  not  be  confused  with  an  enumeration  m^/;?^^/;/^ 
enumerations.  Thus  \\{'F  includes  E'  then  the  items 
comprised  in  E  are  the  same  as  some  of  the  items  com- 
prised in  F,  and  here  E  and  F  comprise  the  same  types 
or  kinds  of  items.  But  if  '  F  comprises  El  then  the 
items  comprised  in  F  will  be  of  a  higher  order  or  type 
than  the  items  comprised  in  E.  Using  the  term  item 
(in  the  first  instance)  to  stand  for  an  entity  of  order 


■'1 


zero,  i.e.,  one  which  is  not  itself  an  enumeration,  an  ^ 
enumeration  comprising  such  items  will  be  of  the  first  ! 
order,  and  an  enumeration  comprising  enumerations  of 
the  first  order  will  be  of  the  second  order ;  and  so  on. 
In  passing  from  enumerations  of  the  first  order,  viz., 
those  which  comprise  mere  items,  to  enumerations  of 
the  second  order  which  comprise  enumerations  of  items, 
we   may  symbolise   the   distinction   by  using   square 
brackets.    Thus  an  enumeration  of  the  first  order  may 
be  illustrated  thus  :  {a,  b,  c,  d,  e,  /  g,  h,  k,  /,  m,  n]  in- 
volving one  size  of  bracket.     Now,  with  these  same 
twelve  items  we  may  illustrate  several  enumerations  of 
the  second  order  which  will  involve  two  sizes  of  brackets 
as  follows: 

[[^,  61   [^,  d,  ej,g\   \_k,  k,  /,  m\   [«]] 

where  the  item-enumerations  are  exclusive  and  four  in 
number. 
Again : 

[L^,  b,  c\   \c,  d,  ej\    [a,/,  h,  k,  /],    \_d,  m,  n~\ 

where  the  item-enumerations  are  not  exclusive  of  one 
another  and  again  are  four  in  number. 
Or  again : 

[[^,  b\   \c,  d,  e\   \bj,  h\   [_d,  k\   [_e,  m,  ;^]] 

where  some  pairs  of  the  item-enumerations  are  exclusive 
and  others  not,  the  total  number  being  five. 

I  n  all  these  illustrations,  a  comma  is  used  to  separate 
the  items  to  be  enumerated  in  constituting  an  enumera- 
tion, and  the  square  brackets  are  used  where  required 
to  indicate  what  is  to  be  taken  as  an  item.     Similarly, 


i  1 


Tl8 


CHAPTER  VIII 


ENUMERATIONS  AND  CLASSES 


119 


using  the  same  twelve  items  we  may  illustrate  an 
enumeration  of  the  third  order ;  which  will  involve  three 
sizes  of  brackets  thus : 

[[[a,  d],    [c,  d,  ^]],     [[<5,/  h\   {d,  /&]],     \[_e,  m,  «]]]  , 

where  there  are  three  items  which  are  enumerations  of 
the  second  order,  of  which  the  first  two  comprises  as 
items  two  first  order  enumerations,  while  the  third  com- 
prises only  one  first  order  enumeration.  Comparing 
our  illustration  of  a  first-order  enumeration  with  the 
first  illustration  of  a  second-order  enumeration,  we  must 
note  that  the  item  n  which  is  of  zero  order  is  to  be  dis- 
tinguished from  the  item  [_n\  which  is  of  the  first  order, 
and  comparing  this  last  with  our  third  order  enumera- 
tion, we  must  distinguish  the  item  [^,  m,  n\  which  is  of 

the  first  order  from    [e,  m,  ft]    which  is  of  the  second 

order.  The  distinction  between  n  and  [n]  is,  therefore, 
that  the  former  is  to  count  as  one  along  with  other  items 
in  constituting  an  enumeration  of  xh^Jirsi:  order,  while 
[fi\  is  to  count  as  one  along  with  other  items  in  consti- 
tuting an  enumeration  of  the  second  order.    Similarly, 

the  distinction  between  [e,  m,  fi\  and  \\e,  m,  n\\ ,  is  that 

the  former  is  to  count  as  one  along  with  other  items 
in  constituting  an  enumeration  of  the  second  or A^x,  while 
the  latter  is  to  count  as  one  along  with  other  items  in 
constituting  an  enumeration  of  the  thij^d  order.  On 
precisely  similar  grounds  we  must  distinguish  between, 

say,  \\a,  b\  \c,  d,  e\\  and      [a,  b\  \c,  d,  e]      ;  for  the 

former  represents  an  enumeration  of  the  second  order 


comprising  ^wo  items  which  are  enumerations  of  the 
first  order,  while  the  latter  represents  an  enumeration 
of  the  third  order  comprising  one  item  of  the  second 
order.  In  the  above  treatment  we  have  in  effect  defined 
the  notions  'item'  and  'enumeration,'  not  as  having  1 
absolute  significance,  but  as  having  relative  significance,  ^ 
in  the  sense  that  the  two  notions  are  indicated  by  the  re- 
lative term  'comprising'  and  its  correlative  'comprised 
in.'  In  other  words  our  proper  topic  has  been  the 
development  of  the  kind  of  relation  expressed  by  the 
verb  comprise. 

Further  to  illustrate  the  principle  that  the  operators  i 
into  and  with  yield  an  enumeration  of  the  same  order 
as  the  enumerations  operated  upon,  we  will  apply  these  ^ 
operators  to  enumerations  of  the  second  order.  When 
abbreviating  the  expression  for  an  enumeration  by  sub- 
stituting a  single  letter  E  or  F,  we  shall  use  as  indices 
I,  2,  3,  ...  to  indicate  the  different  orders  to  which  any 
enumeration  may  belong.    Thus: 

Let  ^'  stand  for  \\_a,  b,  ^],  [cy  d],  \_a,  e,f,g\  [a,  h,  k,  /]J 
and  F''  stand  for  \\a,  b,  c\  \c,  d,  e\  {a,  e,f.g\  \b,  k,  /,  m\\ 

then  theoperation '^'  intoi^''  yields  \^a,  b,  c\  \_a,  e,/,/]j 

and  the  operation  '^'  with  F^'  yields 

[[a,  b,  cl  Ic  dl  [c,  d,  el  [a,  ej,g\  {_aA  k.  /].  \b,  k,  /,  /^]] 

Thus  the  operation  E''  into  F''  yields  C  and  the  opera- 
tion E'  with  F''  yields  H\  where  it  may  be  seen  that 
C  and  H''  stand  for  enumerations  of  the  second  order. 
We  shall  also  require  a  symbol  for  the  result  of  using 
the  operator  into,  where  the  enumerations  are  exclusive 


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121 


of  one  another.    According  as  the  enumeration  yielded 
in  this  case  is  of  the  first,  second,  or  third  order  it  will 

be  symbolised  by  [o],  or  KojJ  or    ["[ojl    .    Thus : 

The  operation  \_a,  b,  c\  into  {d,  e\  yields  [o];  and  the 
operation  \^a,  b,  c\,  \c,  ^J  into  \\a,  d,  e]^  [^,/]]  yields 

[o]  .  Thus  amongst  all  possible  enumerations  of  the 
first  order  we  must  include  [o]  ;  and  amongst  all  those 
of  the  second  order  we  must  include  [o]  .  An  enumera- 
tion characterised  as  being  o  may  be  called  an  empty 
enumeration,  the  symbol  o  having  in  every  case  the 
same  meaning:  the  single,  double,  etc.,  bracket  adds  a 
further  character  to  the  character  zero.  The  symbol  o 
is  obviously  selected  to  indicate  that  the  number  of 
items  in  an  empty  enumeration  is  zero.  As  regards 
items  of  zero  order  none  can  be  called  empty ;  and  there- 
fore none  can  be  symbolised  as  o. 

§  3.    At  this  point  we  must  explain  more  precisely 
the  distinction  between  being  comprised  in  and  being 
included  in.     Thus,  each  of  the  three  items  a,  b,  c  is 
comprised  in  the  enumeration  \a,  b,  c\  and  no  others. 
The  relation  of  comprising  thus  always  correlates  an 
item  or  an  enumeration  of  a  certain  order  with  an  enu- 
meration of  the  next  higher  order.     On  the  other  hand 
j  [a]  is  included  in  [a,  b,  c\  thus  showing  that  the  relation 
j  of  inclusion  is  a  relation  between  enumerations  of  the 
1  same  order. 

The  above  account  suggests  the  elementaryproblem, 
how  many  enumerations  are  included  in  the  enumeration 
[a,  b^  c\  (taken  to  be  an  enumeration  of  the  first  order). 


) 


% 


Since  [o]  being  of  the  first  order  is  included  in  every 
first  order  enumeration,  the  following  first  order  enu- 
merations will  exhaust  all  those  that  are  included  in 
\a,  b,  c\  namely: 

First,  that  which  comprises  no  item :  [o] : 

Secondly,  those  which  comprise  one  item: 
\_d\  and  \b'\  and  \c\ : 

Thirdly,  those  which  comprise  two  items : 
[a,  b~\  and  [a,  c\  and  \by  c\\ 

Fourthly,  that  which  comprises  three  items :  [a,  b,  c\ 
Thus,  within  the  enumeration  \a,  b,  c\  the  number  of 
distinct  enumerations  included  is  8;  for,  in  selecting 
any  enumeration  which  shall  be  included  in  [a,  by  c\ 
we  have,  with  respect  to  each  of  these  three  items, 
the  two  alternatives  of  admitting  or  omitting  it.  Hence 
the  number  of  our  choices  is  2x2x2.  Similarly  in  any 
enumeration  comprising  (say)  n  items,  2"*  enumera- 
tions will  be  included.  Thus,  we  may  write  down  an 
enumeration  of  the  second  order  which  shall  comprise 
all  the  enumerations  of  the  first  order  included  in  [a,  b,  c\ 

Thus :  [[o],  [^],  \b\  {c\  [a,  b-\  [a,  c\  \b,  c\  [a,  b,  ^]] .   In 

general  terms  then:  The  enumeration  of  the  second 
order,  that  shall  comprise  as  its  items  all  the  enu- 
merations of  the  first  order  included  in  a  given  enu- 
meration of  the  first  order  comprising  n  items,  will 
comprise  2^*  items  of  the  first  order. 

§  4.  Having  treated  of  enumerations  we  may  now 
consider  the  relation  between  an  enumeration  and  a 
class.  Whether  a  class  may  or  may  not  be  considered 
as  an  enumeration  of  a  special  kind,  it  will  be  agreed 
that  there  is  involved  in  the  notion  of  a  class  an  element 


122 


CHAPTER  VIII 


ENUMERATIONS  AND  CLASSES 


123 


entirely  absent  from  that  of  a  mere  enumeration.  In 
the  language  of  Mill,  the  denotation  of  a  class  may  be 
said  to  be  determined  by  connotation ;  i.e.  by  a  certain 
1  conjunction  of  adjectives.  But  here  it  is  of  the  utmost 
importance  to  note  that,  on  the  one  hand,  the  substan- 
tival items  constituting  the  denotation  are  united  merely 
by  the  enumerative  '  and ' ;  but,  on  the  other  hand,  the 
adjectival  items  constituting  the  connotation  are  united 
by  the  conjunctional '  and\'  In  fact,  what  is  common  to 
every  logician's  employment  of  the  term  class  is  that 
its  limits  are  determined — not  merely,  if  at  all,  by  a 
mere  enumeration  of  items — but  essentially  by  the 
character  or  conjunction  of  characters  that  can  be  truly 
predicated  of  this  and  of  that  item  that  isto  be  comprised. 
The  distinction  between  an  enumeration  and  a  class  is 
closely  connected  with  that  between  an  extensional  and 
an  intensional  point  of  view,  so  that  logicians  have  con- 
trasted the  extension  of  a  class  with  its  intension  or  the 
intensional  conception  of  a  class  with  its  extensional 
conception.  Phrases  of  this  kind  have  in  fact  been  in- 
troduced in  various  parts  of  this  work;  but  a  more 
direct  way  of  attacking  our  problem  would  be  to  speak 
— not  of  the  extension  and  the  intension  of  a  class — 
but  of  the  extension  of  an  intension  and  of  the  intension 
of  an  extension,  where  the  preposition  ^requires  to  be 
logically  defined.  More  explicitly  I  propose  to  speak 
of  an  intension  as  determining  a  certain  extension,  or 
conversely  of  an  extension  as  being  determined  by  a 
certain  intension.  Thus  the  relation  determining  and 
its  correlative  determined-by  will  indicate  the  required 
connection  and  distinction.    We  shall   not   generally 

^  See  Chapter  III. 


speak  of  an  extension  determining  an  intension  or  of  an 
intension  being  determined-by  an  extension,  but  the 
relation  of  determining  will  be  always  from  the  intension  1 
to  the  extension.    In  short,  it  is  this  direction  of  deter- 1 
mination  which  justifies  Mill's  use  of  the  term   con- 
notationy  and  when  we  are  conceiving  the  converse 
case  of  an  extension  determining  an  intension,  then  we 
may  adopt  for  the  intension  in  this  case  the  convenient 
term  comprehension  as  introduced  by  Dr  Keynes.    The 
word  determining  as  used  above  is  of  course  elliptical. 
In  speaking  of  a  given   intension  or  conjunction  of 
adjectives  as  determining  an  extension,  what  of  course 
is  always  understood  is  that  this  or  that  item  is  or  is 
not  to  be  comprised  in  the  extension  according  as  it  is 
or  is  not  characterised   by  the  given  conjunction  of 
adjectives.     Now  it  will  be  found  that  the  larger  and 
more  familiar  part  of  logical  theory  is  actually  concerned 
— not  with  the  notion  of  extension — but  solely  with 
that  of  intension,  and  that  it  is  only  when  arithmetical 
predicates  come  into  consideration  that  the  notion  of 
extension  seems  to  be  required.    Thus,  taking  the  pro- 
position :  '  Everything  having  the  character  m  has  the 
character/,'  we  may,  for  any  English  letter  standing 
illustrativelyforan  adjective,  introduce  the  corresponding 
Greek  letter  in  a  purely  symbolic  sense  to  stand  for  the 
class  determined  by  that  adjective.    Thus  the  above 
distributively  expressed  proposition  may  be  rendered: 
*  the  class  /x  is  included  in  the  class  tt!    Again,  if  we 
conjoin  with  the  above  proposition :  '  Everything  having 
the  character/  has  the  character  m'  we  reach  the  form 
of  proposition :  '  The  class  //,  coincides  with  the  class 
ttI  Now  the  relation  coincides  is  analogous  to  the  relation 


124 


CHAPTER  VIII 


ENUMERATIONS  AND  CLASSES 


125 


of  co-implication,  in  that  both  are  transitive,  symmetrical 
and  reflexive;  i.e.,  they  have  the  properties  of  equi- 
valence or  identity.  In  this  way  we  may  speak  of  the 
identity  of  a  class  determined  by  one  adjective  with 
that  determined  by  another  (merely  as  expressing  a 
symbolic  or  abbreviated  formula)  without  implying  that 
there  is  any  real  entity  to  be  called  an  extension  or  a 
class  to  which  the  strict  relation  of  identity  could  be 
applied.  All  this  is  assumed  in  the  next  chapter,  where 
we  shall  represent  the  force  of  propositions  by  means 
of  closed  figures.  In  spite  then  of  the  prominent  em- 
ployment of  the  word  class  both  in  the  treatment  of 
propositions  and  still  more  in  that  of  the  principles  of 
syllogism,  it  may  be  maintained  that  there  is  no  real 
reference  in  thought  to  the  class  as  an  extension,  but 
only  a  figurative  or  metaphorical  application  of  the 
word  which  serves  to  bring  out  certain  analogies  between 
such  notions  as  inclusion,  exclusion,  and  exhaustion 
which  apply  primarily  to  parts  and  wholes  and  are 
transferred  as  relations  between  propositions  and  their 
constituent  elements.  Some  logicians  have  even  gone 
so  far  as  to  say  that  the  spatial  relations  amongst  plane 
closed  figures  represent  the  actual  mode  of  thought  by 
means  of  which  we  are  able  to  comprehend  logical 
.  relations.  I,  however,  reject  this  extreme  point  of  view, 
but  suggest  that  the  mere  fact  that  we  are  able  to 
represent  logical  relations  by  analogy  with  relations 
amongst  spatial  figures  almost  justifies  our  maintaining 
that  the  idea  of  an  extension  determined  by  an  intension 
>is  a  logically  valid  concept.  The  full  significance  of 
such  a  scheme  as  Eulers  diagrams  for  representing 
class-relationships  has,  in  my  view,  been  inadequately 


!l. 


recognised.    It  should  be  pointed  out  that  the  boundary 
line  of  a  closed  figure  may  be  taken  as  the  proper  ana- 
logue of  the  intension,  while  the  area  within  that  boundary 
is  the  proper  analogue  of  the  extension.  This  suggestion 
brings  out  the  following  analogies:  firstly,  that   it   is 
intension  which  determines  extension  in  the  same  way 
as  a  boundary  line  determines  the  enclosed  area  and 
separates  this  area  from  the  remaining  area  outside; 
secondly,  that  we  can  apprehend  in  thought  the  full 
determining  intension  in  the  same  way  as  we  can  op- 
tically grasp  the  single  boundary  in  its  entirety;  and 
thirdly,  that  in  general  we  cannot  in  thought  enumerate  li 
all  the  items  which  are  to  be  comprised  in  the  extension, 
just  as  we  cannot  exhaustively  present  to  the  eye  the  ^t 
several  and  innumerable  points  within  the  given  enclosed 
area.    On  the  other  hand,  though  the  several  points 
cannot  be  exhaustively  presented  to  the  eye  and  yet 
the  area  presents  itself  ocularly  as  a  unitary  whole, 
similarly  it  would  seem   that   though  we  cannot  ex- 
haustively enumerate  in  thought  the  members  of  a  class 
yet  we  can  conceive  the  class  or  rather  the  extension 
as  a  unitary  whole.    Again  we  may  make  within  the 
area  actual  dots  of  a  finite  number  which  thus  constitute  " 
a  literal  (though  of  course  not  exhaustive)  enumeration, 
and  thus  the  force  of  the  diagram,  as  providing  analogues 
to  logical  relations,  is  still  further  brought  out,  in  that 
we  may  think  one  by  one  of  the  objects  which  we  have 
selected — not  arbitrarily — but  on  the  ground  that  each 
of  them  is  actually  characterised  by  the  adjectives  which 
determine  the  class.    Whether  this  analogy  between  a 
psychical  image  or  perception  of  an  area  and  the  logical 
conception  of  a  class,  justifies  our  regarding  the  latter  as 


/I. 


i. 


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CHAPTER  VIII 


ENUMERATIONS  AND  CLASSES 


127 


a  genuine  concept,  is  a  debatable  psychological  problem. 
I  Dismissing  this  problem  we  must  return  to  the  strictly 
logical  question  whether  a  class  is  a  genuine  entity. 

§  5.    If  we  provisionally  allow  a  class  comprised  of 

individual  existents  to  be  an  existent,  then  as  an  existent 

0        it  is  of  a  different  order  from  any  individual  comprised 

in  it.    Similarly  a  class  comprising  adjectives — if  it  is  to 

be  called  an    adjective — must   be  an   adjective  of  a 

{U^  yU^i^    different    order  from  any   adjective  comprised    in   it. 


/ 


Similarly  for  a  class  of  propositions.  Therefore  the 
question  whether  a  class  is  a  genuine  entity  admits  at 
any  rate  that  it  is  not  of  the  same  order  of  being  3S  any 
item  which  it  comprises.  Whether  this  or  that  is  a 
genuine  entity  can  only  be  answered  when  we  have 
provided  a  test  of  genuineness.  The  only  general  test 
which  I  can  conceive  of  is  as  to  whether  the  entity 
intended  to  be  meant  (in  using  such  a  word  as  class) 
can  serve  as  subject  of  which  some  predicate  can  be 
truly  asserted^  Thus,  as  an  illustration  of  the  general 
question,  we  may  ask  whether  a  proposition  is  a  genuine 
entity,  and  taking  the  proposition  matter  exists,  the  reply 
would  be  in  the  affirmative,  inasmuch  as  we  can  make  the 
assertion  'that  matter  exists  was  rejected  by  Berkeley.' 
Similarly  with  regard  to  the  genuineness  of  a  'class' 
which  is  the  topic  under  consideration.  Taking  for 
example  the  class  apostles,  we  may  assert  'that  this  class 
numbers  twelve.'  Inasmuch  as  this  statement  will  be 
admitted  to  be  true,  the  only  relevant  question  that 
could  arise  would  be  as  to  whether  the  number  twelve 
is  predicated — not  of  the  kind  of  entity  called  a  class — 
but  rather  of  its  determining  adjective.  To  this  it  may 
^  Cf.  the  treatment  of  *  6"  is '  in  Chapter  V. 


be  replied  that  since  different  intensions  may  determine 
the  same  class,  the  number  predicated  cannot  be  said 
to  be  predicated  of  one  of  these  determining  intensions 
rather  than  of  any  other.    In  other  words,  we  may  say 
that  the  adjective  twelve  is  of  such  a  kind  that,  taking 
any  two  co-implicative  determining  intensions,  if  it  can 
be  truly  predicated  of  one  it  can  be  also  truly  predicated 
of  the  other.     Since  then  it  is  indifferent  of  which  of 
these  several  co-implicatives  the  adjective  twelve  is  to 
be  predicated,  it  seems  to  follow  that  it  is  not  predicated 
of  any  of  them  whatever.    Let  us  take  /  and  q  to  stand 
for  two  co-implicative  intensions.     This  means  more 
precisely  that,  if  anything  is  characterised  as  being/, 
it  will  be  characterised  also  as  being  q ;  and  conversely, 
if  anything  is  characterised  as  being  q,  it  will  be  also 
characterised  as  being/.   It  is  then  clear  that  the  relation 
called  co-implication  is  both  symmetrical  and  transitive. 
Let  us  then  assume  the  question  at  issue,  namely  that 
there  is  such  an  entity  as  a  class.    Then  our  conception 
of  a  class  involves  the  universal  statement  that  any 
given  intension  determines  one  and  only  one  class.    In 
this  way  the  relation  of  co-implication  subsisting  between 
/  and  q,  may  be  resolved  into  the  statement  that/  de- 
termines a  certain  unique  entity  which  is  the  same  as 
that  which  is  determined  by  q:  that  entity  being  what 
we  have  taken  to  be  a  class.     Here  the  relation  deter- 
mining, which  relates  an  intension  to  its  extension,  is 
what  is  called  a  many-one  relation,  because  there  may 
be  many  different  intensions  which  determine  a  single 
extension.    Similarly,  the  relation  determined-by ,  which 
relates  an  extension  to  its  intension,  is  what  is  called  a 
one-many  relation,  because  there  is  only  one  extension 
which  is  determined  by  many  different  intensions. 


128 


CHAPTER  VIII 


ENUMERATIONS  AND  CLASSES 


129 


§  6.  We  may  generalise  the  above  by  taking,  as  a 
typical  illustration,  the  relation  of  a  man  to  the  country 
which  he  inhabits.  Since  there  is  only  one  country  which 
a  man  inhabits  the  relation  inhabits  is  many-one,  and 
its  converse  inhabited-by  is  one-many.  Now  let  us 
combine  these  two  correlatives  in  the  following  form :  'A 
inhabits  the  country  that  is  inhabited  by  B'  The  relation 
thus  constructed  of  A  to  B  (which  may  be  called  the  rela- 
tion compatriot-of)  is  obviously  symmetrical  and  tran- 

di  sitive.  It  is  symmetrical  because  there  is  only  one  country 
which  A  can  inhabit  and  one  country  only  which  B  can 
inhabit,  so  that  to  say  that  A  and  B  inhabit  the  same 
country  exhibits  a  symmetrical  relation,  since  the  terms 

y  A  and  B  may  be  interchanged.  Again,  the  relation  is 
transitive  since  if  A  and  B  inhabit  the  same  country, 
and  B  and  C  also  inhabit  the  same  country,  it  follows, 
since  no  one  inhabits  more  than  one  country,  that  A 
and  C  inhabit  the  same  country.  The  identity  of  the 
country  inhabited  by  this,  that  and  another  man  is  a 
symmetrical  and  transitive  relation,  and  it  is  upon  these 
properties  of  identity  that  the  symmetry  and  transitive- 
ness  of  the  relation  compatriot-of  d^p^nds.  In  fact,  to 
speak  of  the  country  inhabited  by  A  would  not  be  a 
legitimate  expression  unless  there  was  one  and  only  one 
country  which  a  man  could  be  said  to  inhabit.  Thus 
it  will  be  seen  that,  when  we  combine  in  this  sort  of 
way  any  many-one  relation  with  its  correlative  (which 
is  necessarily  one-many),  then  we  have  constructed  a 
relation  which  has  the  two  properties  transitive  and 
symmetrical.  But  the  reverse  of  this  does  not  obviously 
hold — that  is,  given  a  transitive  and  symmetrical  relation 
it  does  not  obviously  follow  that  this  relation  can  be 
resolved  into  a  combination  of  a  certain   many-one 


relation  with  its  correlative.  If,  however,  this  converse 
proposition  may  be  taken  as  axiomatic,  it  would  follow, 
from  the  two  properties — symmetrical  and  transitive — 
which  hold  of  co-implication  between  two  intensions/ 
and  q,  that  there  is  a  certain  thing  which/  determines 
and  which  is  determined  by  q.  To  this  kind  of  entity 
we  apply  the  name  class.  It  is  therefore  only  by 
assuming  the  theorem  that  any  given  relation  that  is 
symmetrical  and  transitive  can  be  resolved  as  above  in 
terms  of  a  many-one  relation  and  its  converse  that  we 
cd^n  prove  that  the  notion  of  a  class  represents  a  genuine 
entity.  But,  in  making  use  of  the  analogy  between 
(a)  'the  country  inhabited  by  any  of  the  men  that  are 
compatriots  of  one  another'  and  (b)  'the  class  determined 
by  any  of  the  intensions  that  are  co-implicants  of  one 
another,'  it  must  be  further  pointed  out  that,  just  as  the 
'country'  under  consideration  is  not  the  same  as  the 
compatriots  taken  in  their  totality,  so  the  'class'  under 
consideration  is  not  the  same  as  the  co-implicants  taken 
in  their  totality.  Since,  then,  the  'compatriots'  or  the 
'co-implicants'  taken  in  their  totality  would  constitute  a 
class,  the  attempt  to  prove  the  above  theorem  would 
entail  a  petitio  principii,  when  applied  to  the  question 
of  the  genuineness  of  the  notion  'class.'  Merely  from 
the  symmetry  and  transitiveness  of  the  relation  {s) 
compatriot,  we  cannot  prove  that  there  is  a  certain 
many-one  relation  {r)  inhabit,  or  (an  entity  x)  viz.  a 
certain  country  distinct  from  the  class  of  compatriots  all 
of  whom  inhabit  (r)  that  country  {x\  Our  certainty  that 
'  when  a  is  ?  to  b,  then  a  is  r  tox  and  xisrto  b  for  some 
x'  is  due  to  the  fact  that,  in  order  to  construct  any  s,  we 
must  first  have  been  given  r. 

J.  L.  9 


130 


X  CHAPTER  IX 

THE  GENERAL  PROPOSITION  AND  ITS 
IMMEDIATE  IMPLICATIONS^ 

§  I.  We  may  define  a  general  proposition  as  one  in 
which  the  subject  is  constructed  by  prefixing  an  appli- 
cative to  a  general  name.  According  to  this  definition, 
the  only  kind  of  proposition  which  is  not  general  would 
be  that  in  which  the  subject  is  expressed  by  a  proper 
name;  and  the  general  proposition  would  include  two 
forms  of  singular  proposition :  namely,  where  the  general 
subject-term  is  prefixed  by  'a  certain'  or  by  *the.' 

We  must  first  point  out  that  there  are  what  may  be 
csWed  pure  general  propositions,  where  the  general  term 
is  represented  by  the  word  'thing'  in  its  absolute  univer- 
sality: for  example  *  Everything  is  finite,'  'some  things 
are  extended.'  In  these  propositions  the  subject-term  has 
merely  substantival  without  any  adjectival  significance. 
But  ordinary  propositions  in  every-day  use  apply  to 
subjects  adjectivally  restricted;  in  other  words,  there  is 
an  adjectival  significance  in  the  subject-term  as  well  as 
in  the  predicate  term. 

§  2.  Using  the  capital  letters  P  and  Q  for  general 
or  class  terms  and  the  corresponding  small  letters/ and  ^ 
to  represent  their  adjectival  significance,  then  the  correct 

^  This  Chapter  should  be  read  in  close  connection  with  Chapter  III. 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    131 

expression  for  the  affirmatives,  universal  and  particular, 
would  be: 

Every  P  is  ^, 

Some  P  is  ^, 

which  brings  out  the  substantival  significance  in  the 
subject-term  and  the  adjectival  significance  of  the  predi- 
cate term.  Now  these  forms  are  at  once  seen  to  be 
equivalent  respectively  to 

A.   Everything  that  is/  is  ^, 
/.    Something  that  is/  is  ^\ 

where  the  adjective  /  occurs  as  predicate  in  the  sub- 
ordinate clause,  and  the  introduction  of  the  word  'thing' 
indicates  an  ultimate  reference  to  the  absolutely  general 
substantive.  The  negative  general  categoricals  may  first 
be  expressed  in  the  forms : 

Everything  that  is/  is-not  q, 
Something  that  is  /  is-not  ^, 

where,  in  attaching  the  negative  to  the  copula,  it  is  to 
be  understood  that  the  adjective  q  is  denied,  in  the  first 
case,  of  everything  that  is  /;  and,  in  the  second  case, 
of  something  that  is  /.  But  these  negatives  are  ex- 
pressed less  am^biguously  as  the  contradictories  of  /  and 
A  respectively,  thus: 

E,    Nothing  that  is/  is  q^ 

O.    Not-everything  that  is/  is  q, 

^  The  subject-terms  in  these  two  readings  are  often  contrasted  as 
being  the  one  in  denotation  and  the  other  in  connotation.  This 
disregards  the  fact  that  the  meaning  of  the  term  P  contains  a  con- 
notative  as  well  as  a  denotative  factor,  while  the  phrase  *  thing  that 
is  /'  contains  a  denotative  as  well  as  a  connotative  factor.  Hence 
the  two  readings  are  not  properly  to  be  contrasted. 

9—2 


132 


CHAPTER  IX 


where  the  negative  is  prefixed  to  the  propositions  as  a 
whole,  and  must  not  be  falsely  supposed  to  qualify  the 
subject-term. 

By  expressing  the  propositions  A,  /,  E,  O,  in  the 
above  forms  the  true  logical  nature  both  of  obversion 
and  of  conversion  can  be  explained.  Thus  the  negative 
that  is  introduced  or  omitted  by  the  process  of  obversion, 
is  to  be  attached  to  the  adjectival  factor  alone  in  the 
predicate ;  and  hence  the  four  propositions : 

A,    Every 


-thing  that  \^  p\s  q 


E.    No 

/.      Some 

O,     Not-every, 

become  (respectively)  by  obversion 

E,,  No 

Ai.  Every        .      .  .        i    .  •    .  • 

^        V  -thms:  that  is  p  is  non-^ 

O,.   Not-every^  ^ 

/^.     Some 

where  E^,,  Aj,,  (9^,  7^,  are  of  the  forms  E,  A,  O,  /, 
but  having  in  each  case  non-^?-  for  predicate  in  place 
of  q.  Here  the  suffix  b  indicates  that  the  proposition 
is  obtained  by  obversion.  Applying  again  the  process 
of  obversion  to  Ei,,  A^,,  (9^,  h,  we  obtain 


-  -thing  that  is  /  is  non-non-^. 


Ai,!,.  Every 

E,,,  No 

/j^.  Some 

Obb.  Not-everyj 

Now,  by  the  principle  of  double  negation,  non-non-^ 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    133 

is  equivalent  to  q.  Hence  ^33,  ^33,  Z^^,  (9^,  are  re- 
spectively equivalent  to  A,  E,  /,  O — showing  that  the 
propositions  obtained  by  obversion  are  equipollent,  i.e. 
formally  coimplicant. 

The  process  of  simple  conversion  may  be  exhibited 
by  interpolating  intermediate  steps,  where  first  the  ad- 
jectival factor  in  the  subject  is  removed  to  the  predi- 
cate; then,  by  the  commutative  law  for  conjunctives, 
the  adjectives  are  transposed;  and  finally,  the  first  ad- 
jectival factor  in  the  predicate  is  removed  back  to  the 
subject: 

E.    Nothing  that  is/  is  ^ 

=  Nothing  is/  and  q 

=  Nothing  is  q  and/ 

=  Nothing  that  is  q  is  p. 

/.     Something  that  is/  is  ^ 

=  Something  is/  and  q 

=  Something  is  q  and/ 

=  Something  that  is  q  is/. 

The  forms  A  and  (9,  not  being  directly  convertible, 
must  first  be  obverted  and  then  converted,  giving  the 
contrapositive,  i.e.  the  converted  obverse,  thus: 

A.    Everything  that  is/  is  q 

=  Nothing  that  is  /  is  non-q 

=  Nothing  that  is  non-^  is/. 

O.    Not  everything  that  is/  is  ^ 

=  Something  that  is/  is  non-^ 

=  Something  that  is  non-^  is/. 

We  have  so  far  taken  E,'No  P  is  q'  to  be  equiva- 
lent to  'Nothing  is  pq';  and  /,  'Some  P  is  q'  to  be 
equivalent  to  'Something  is /^.'    In  other  words  we 


134 


CHAPTER  IX 


t 


have  reformulated  the  E  and  /  propositions,  in  which 
the  subject-term  is  restricted  by  the  adjective/,  by  using 
as  subject-term  the  unrestricted  reference  expressed  by 
the  word  'thing.'  In  the  same  way,  A,  'Every  P  \s  q' 
and  (9,  'Not-every  P  is  q'  may  be  reformulated  thus: 
Ay  'Everything  is  /  or  ^';  and  O,  'Not-every thing  is 
p  or  q  \  since  '/  or  q  is  equivalent  to  'q  if/.'  In  these 
reformulations,  the  form  of  the  proposition  as  universal 
or  particular  and  as  negative  or  affirmative  is  unaltered : 
the  ^^'-proposition  is  expressed  as  an  ^-proposition,  the 
/  as  an  /,  the  A  as  an  A,  the  O  as  an  O.  We  have,  in 
I  short,  merely  reduced  all  the  propositions  to  the  same 
common  denominator  (so  to  speak)  by  using  the  nar- 
rowest reference  for  our  subject-term  which  will  be  suf- 
ficiently wide  to  include  all  the  subject-terms  that  may 
occur  in  any  given  connected  system  of  statements. 
This  kind  of  transformation  has  been  called  'existential,' 
but  since  the  term  'existential'  has  been  so  persistently 
misunderstood,  it  will  be  preferable  to  speak  of  'instan- 
tial'  instead  of 'existential'  formulation.  In  this  mode 
of  formulation,  a  further  question  arises,  namely  that  of 
interpretation-,  in  particular,  as  to  whether  the  propo- 
sition— given  to  be  reformulated — is  to  be  understood 
to  include  (implicitly  or  explicitly)  the  statement  that 
there  are  instances  characterised  by  /,  where  /  is  the 
adjective  connoted  by  the  subject-term:  i.e.  to  include 
the  affirmative  statement  'Something  is/.'  Here  it  is 
to  be  observed  that  the  instantial  statement  'Some- 
thing is/'  is  implicitly  contained  in  'Something  is  pq^ 
but  not  in  'Nothing  is/^.' 

§3.  We  will  distinguish  the  proposition  which  ^^«/^m^ 
the  instantial  affirmation  of  its  subject-adjective  from 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    135 

the  (otherwise)  same  proposition  which  does  not  contain 
this  affirmation  by  using  the  suffix/"  for  the  former,  and 
n  for  the  latter.  In  order  to  change  (say)  T„  into  7}, 
where  7"stands  for  any  proposition  having  an  adjectival 
factor  (say  /)  in  its  subject-term,  we  must  add  to  T^ 
the  statement  'Something  is  /,'  i.e.  we  must  conjunc- 
tively combine  with  T^  the  instantial  affirmation.  And, 
in  order  to  change  (say)  Tf  into  T^,  we  must  subtract 
from  7)rthe  statement  'Something  is/,'  i.e.  we  must  al- 
ternatively combine  with  7) the  instantial  denial.  Thus : 
given  T^,  7) will  be  expressed  '  T^  ^«<^ Something  is/'; 
given  7),  T^  will  be  expressed  '  Tyor  Nothing  is/.' 

Applying  these  expressions  to  the  forms ^„,  I/,A„,  Of 
we  have: 

En  means  'Nothing  is  pq.^ 
If  „  'Something  is /^.' 
An  „  'Nothing  is  pq.^ 
Oj      „       'Something  is /^.' 

Ef  means  'Nothing  \^ pq  d!«</ Something  is/.' 

/„       „       'Something  \s pq  or  Nothing  is    /.' 

A/      „       'Nothing  is  pq  and  Something  is/.' 

On      )i       'Something  is /^  <7r  Nothing  is    /.' 
r 

From  the  meanings  of  the  symbols,  as  thus  shown, 
the  following  rules  will  be  evident : 

{a)  for  obtaining  the  co-opponent:  interchange/" 
with  n  and  A  with  0\  or /"  with  n  and  E  with  /  (with- 
out any  other  change) ; 

(b)  for  obtaining  a  sub-implicant;  change  either/" 
into  n,  or  A  into  /,  or  E  into  O  (without  any  other 
change). 

Since  super-implication  is  the  reverse  of  sub-impli- 
cation, rule  {b)  reversed  shows  how  to  obtain  a  super- 


136 


CHAPTER  IX 


impHcant.  Moreover,  since  a  sub-opponent  is  a  sub-im- 
plicant  of  the  co-opponent  and  a  super-opponent  is  a 
super-implicant  of  the  co-opponent,  rules  (a)  and  (d) 
combined  show  how  to  obtain  a  sub-opponent  or  super- 
opponent.  Lastly,  since  a  sub-implicant  of  a  sub-im- 
plicant  is  a  sub-implicant,  and  a  super-implicant  of  a 
super-implicant  is  a  super-implicant,  all  the  several  sub- 
implicants,  super-implicants,  sub-opponents  or  super- 
opponents  can  be  found.  In  the  following  mnemonic 
diagram,  an  arrow  is  used  to  point  from  a  super-impli- 
cant to  a  sub-implicant,  and  co-opponents  are  placed 
diagonally  opposite.  The  diagram  is  required  in  place 
of  the  ordinary  square  of  implication  and  opposition, 
because  of  the  distinction  introduced  between  two  pos- 
sible interpretations  of  A,  B,  /,  or  O. 


Here, 

Ey^y  E„y  0}  are  super-implicants  of  0„,  and  are  .*.  super-opponents  of -^y, 
Ay^y  An,  1/  are  super-implicants  of  /„,  and  are  /.  super-opponents  of  jE^, 
/«,  Ify  An  are  sub-implicants  of  A/^  and  are  .*.  sub-opponents  of  0„, 
Ont  Of,  En  are  sub-implicants  of  E/^  and  are  .*.    sub-opponents  of    /„. 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    137 

But,  taking  the  laterally  outstanding  rectangle  A^,  E^, 
Ofy  If,  it  must  be  observed  that  no  relation  of  implica- 
tion or  opposition  holds  of  A^  to  E,, ,  of  E„  to  Of,  of 
Oflo  Ify  or  of  Ify  to  A^\  i.e.  the  sides  of  this  rectangle 
exhibit  the  relation  of  independence. 

The  general  nature  of  these  results  is  that  where 
any  proposition  is  interpreted  as  having  less  determin- 
ate significance,  it  will  be  a  super-implicant  or  super- 
opponent  of  fewer  propositions  and  a  sub-implicant 
or  sub-opponent  of  a  larger  number.  Thus  Aj-  is  su- 
per-implicant to  A,,  and  ly  and  /„,  but  A^  is  super- 
implicant  only  to  /„;  and  Ay  is  super-opponent  to  Ef 
and  E^  and  Of,  but  A„  (being  sub-opponent  to  O^  is 
super-opponent  to  none  of  the  propositions  in  the  octa- 
gon. Conversely  where  any  proposition  is  interpreted 
as  having  more  determinate  significance,  it  will  be 
sub-implicant  or  sub-opponent  to  fewer  propositions 
and  super-implicant  or  super-opponent  to  a  larger 
number. 

Similar  modifications  of  the  traditional  scheme  are 
required  for  inferences  involving  conversion.  It  will  be 
found  that  equipollent  conversion  holds  for  E^  and  If 
but  not  for  ^y  and  /„;  and  that  sub-altern  conversion 
holds  in  passing  to  /  from  ^y  but  not  from  A^\  and  in 
passing  from  E  to  O^  but  not  to  Of, 

Since  each  of  the  propositions  A,  E,  /,  O  can  be 
interpreted  in  two  ways,  there  are  several  possible 
schemes  of  interpretation  of  the  four  together,  in  accord- 
ance with  which  a  square  can  be  extracted  from  the 
above  octagon.  Of  these  combinations,  the  following 
are  the  most  reasonable. 


1/ 


independent 

independent 


Or 


13S  CHAPTER  IX 

(i)   A„EJ^O,, 

Here  the  universals  are  interpreted  as  containing 

no  instantial  affirmation,  while  the  par- 
ticulars implicitly  contain  instantial 
affirmation  of  the  subject-term.  This 
is  the  simplest  interpretation,  since 
each  proposition  is  expressed  as  un- 
compounded,  the  universals  being 
merely  instantial  denials,  and  the  par- 
ticulars, merely  instantial  affirmations. 

(2)  A,E,I,0,. 

Here  all  the  four  propositions  are 
interpreted  as  containing  instantial 
affirmation  of  the  subject-term;  so 
that  the  universals  have  to  be  ex- 
pressed in  a  compound  form. 

(3)  A„EJ„0„, 

Here  all  the  four  propositions  are 
interpreted  as  containing  no  instantial 
affirmation,  so  that  the  particulars 
have  to  be  expressed  in  a  compound 
form. 

This  is  the  reverse  of  the  first 
interpretation,  each  proposition  being 
expressed  as  a  compound:  the  uni- 
versals containing  instantial  affirma- 
tion of  the  subject-term,  and  the 
particulars  containing  no  instantial 
affirmation. 


(5)   A,EJ,0^. 

Here  the  affirmatives  contain  in- 
stantial affirmation  of  the  subject- 
term,  and  the  negatives  contain  no 
instantial  affirmation. 


Ib     sub-  contrary     0„ 


sub-  contrary 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    139 

§  4.  These  five  selected  schemes  must  be  compared 
with  the  traditional  doctrine  on  the  relationsof  y^,^,/,  O, 
Special  explanation  is  required  to  justify  the  application 
of  the  terms  contrary,  contradictory,  etc.,  in  the  tradi- 
tional scheme,  where  all  the  four  general  propositions 
are  to  be  understood  to  assume  that  there  are  instances 
of  the  subject-term.  This  interpretation  differs  from 
interpretation  (2)  given  above,  where  each  proposition 
is  interpreted  as  containing  this  affirmation;  for,  incase 
there  were  no  instances  of  the  subject-term,  either  of 
the  four  forms  of  proposition  would,  on  the  traditional 
scheme  be  meaningless,  whereas  on  interpretation  (2) 
they  would  be  false.  Thus  ly  and  Of  would  both  be 
false,  i.e.  A^  and  E^  would  both  be  true,  supposing 
there  to  be  no  instances  of  the  subject-term ;  whereas 
the  traditional  scheme,  precluding  this  possibility  from 
the  outset,  asserts  that  /  and  O  cannot  both  be  false, 
i.e.  that  A  and  E  cannot  both  be  true.  This  presuppo- 
sition of  traditional  logic  is  concealed  from  the  ordinary 
reader  by  the  universal  employment  of  Euler's  diagrams, 
in  which  the  subject-class  is  indicated  by  an  actual 
circle,  so  that  the  limiting  case,  where  the  class  vanishes, 
is  never  represented.  Furthermore,  the  conversions 
authorised  on  the  traditional  scheme  derive  their  validity 
from  the  assumption  that  there  are  instances  not  only 
of/,  but  also  of  q  and  of  non-/  and  of  non-^,  where/ 
and  q  are  the  adjectival  factors  in  the  subject  and  predi- 
cate terms.  These  assumptions  again  are  tacitly  involved 
in  Euler  s  diagrams,  where  the  circles  for  P  and  Q  are 
not  allowed  to  vanish  or  to  exhaust  the  universe. 

§  5.  In  what  follows  we  shall  adopt  the  traditional 
view  that  there  are  instances  of/,  of  non-/,  of  q^  and  of 


140 


CHAPTER  IX 


non-^;  and  on  this  assumption  we  proceed  to  consider 
all  the  formal  relations  amongst  the  propositions  in- 
volving /  or  non-p  with  ^  or  non-^.  The  symbols 
A\  E\  I\  O'  will  stand  respectively  for  the  propositions 
A,  E,  /,  O  when  modified  by  negating  both  the  subject 
and  the  predicate  adjective.  Thus,  using  the  following 
abbreviative  substitutions,  viz.,  'All'  for  *  every,'  '/'  for 
'thing  that  is/,'  and  '/'  for  non-/,  we  have 


A=  All  p'\sq 
E=  No  pis  q 
I  =  Some  p'\s  q 
(9=  Not  all/  is  q 


A'=-  All  p'lsq 
E  ^  No  ^  is  ^ 
/'  =  Some  pis  q 
a  =  l^ot  2l\  p  IS  q 


giving  a  list  of  eight  distinct  (i.e.  non-equipollent) 
general  categoricals,  as  an  extension  of  the  usual  four. 
Adding  to  this  list  the  propositions  whose  subject  and 
predicate  terms  are  p-q,  p-q,  q-p,  q-p,  q-p,  q-p  we  obtain 
in  all  32  categoricals;  of  non-equipollents,  however, 
there  are  only  32  h- 4,  since  by  means  of  obversion  and 
simple  conversion,  each  proposition  may  be  expressed 
in  four  equipollent  forms.    This  is  shown  in 


By  Obversion 
(i)  .1-* 


Table  I 

By  Conversion 
(ii)        -^        (iii) 


By  Obversion 

(iv) 


A 
A' 
E 
E 


All 
All 
All 
All 


P 
P 
P 
P 


No 


s  q—    No    /  is  q  — 

s  q—   No   /  is  q=  No 

s  ^=   No   p  is  q=  No 

s  ^=    No   p  is  q—  No 


^is^: 

q\sp 

q\sp- 


q'lsp 

0  Not  all/  is  ^  =  Some/  is  ^  =  Some  q  isp 
Cf  Not  all/  is  ^— Some/  is  ^  =  Some  q\sp 

1  Not  all/ is  ^  =  Some/ is  ^= Some  ^  is/ 
/'  Not  all/  is  ^  =  Some/  is  ^  =  Some  q'\sp 


Al 
Al 
Al 
Al 


^is/ 
q\sp 
qisp 
q'lsp 


=  Not  all  ^  is  / 
=  Not  all  q  isp 
=  Not  all  q  isp 
=  Not  all  q'lsp 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    141 

In  this  table  column  (i)  gives  propositions  of  the 
form  A  or  O  (which  admit  of  simple  contraposition); 
the  second  column  is  derived  by  obversion,  giving  pro- 
positions of  the  form  E  or  I  (which  admit  of  simple 
conversion)  ;  the  third  is  next  derived  by  (simple)  con- 
version, giving  again  propositions  of  the  form  E  or  /; 
and  the  fourth  is  derived  again  by  obversion,  giving 
propositions  of  the  form  A   or  O.    The  processes  of  \ 
obversion  and  simple  conversion  being  reciprocal,  give 
equipollents,  i.e.  formal  co-implicants.    The  relation  of ' 
contradiction  (i.e.  formal  co-opposition)  is  seen  in  the 
first  instance  from  the  predesignations  'All'  versus  'Not 
air  and  'Some'  versus  *No,'  and,   on  account  of  the 
equipoUences  tabulated,  each  of  the  four  universal  pro- 
positions in  any  the  same  row  is  related  as  contradictory 
to  each  of  those  in  the  ordinally  corresponding  row  of 
particulars:  i.e.  A  to  O,  A'  to  0\  E  to  /,  E'  to  I\ 

Table  II 


'erse 


...  f 


All  /  is  ^ 

...  \    No  p'lsq 

{No  ^  is/ 

All  q'\sp 

i-. 
Inverse f  Some/ is  ^ 


[Obverse . . . 
IContrapositive 


Obverted  Inverse 
[Obverted  Converse 
Converse 


^ Not  all/  \sq 


...  \^ 


Not  all  q  is/ 

Some  q  is/ 

t 
All    /  is  ^ 


E 

No    p'\s  q 

All     /  is  ^ 

\  . 

Some  q  \sp 
Not  all  q  is/ 


Not  all/ is  q 

Some  /  is  ^ 

All     q  is/ 

I  . 

No     q\sp 
No    p'\s  q 


Some  /  is  ^ 
Not  all/ is  ^ 


Not  all  q  is/ 
Some  q\sp 
Some  /  is  ^ 


O 
Not  all/  is  a 

+ 

Some  /  is  ^ 

♦ 

Some  ^is^ 
Not  alibis/ 


Not  all/  is  q 

Table  II   gives  all  the  implications  amongst  the 
given  propositions  that  can  be  drawn  by  alternate  ob- 


142 


CHAPTER  IX 


version  and  conversion,  beginning  first  with  obversion, 
and  next  with  conversion.  The  arrows  in  the  table 
show  the  direction  of  the  inference,  and  where  there  is 
no  arrow  there  is  no  inference.  This  table  includes  the 
equipollents  of  Table  I ;  and  contains  also  the  sub-altern 
conversions  which  are  required  for  the  form  A,  When 
a  proposition  of  the  form  O  is  to  be  converted  the 
process  must  stop.  The  same  table  could  be  written  out 
iox  A'.E^Py  O' .  Having  in  this  way  given  exhaustively 
the  relations  of  equipollence,  contradiction,  sub-  and 
super-implication  and  opposition,  it  remains  to  deal 
with  the  relation  oi  independence.  This  will  hold  between 
the  following  pairs:  A  and  A\  E  and  E\  A  and  Cf, 
A^  and  (9,  E  and  /',  E'  and  /,  for  which  we  shall  use 
the  technical  terms  complementary  for  independence 
between  universals,  sub-complementary  for  independ- 
ence between  particulars,  and  contra-complementary  for 
independence  between  a  universal  and  a  particular. 

All  these  results  are  expressed  in    the    following 
^octagon  of  implication  and  opposition. 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    143 

§  6.  The  above  account  of  the  processes  of  sub-alter- 
nation, obversion  and  conversion  of  general  categorical 
propositions  is  based  upon  the  logical  relations  amongst 
compound  propositions  explained  in  Chapter  III,  by 
applying  these  latter  without  modification  in  the  form 
of  logical  relations  amongst  adjectives.  Now  this  corre- 
spondence may  be  further  developed  by  bringing  out 
the  analogies  between  the  universal  and  particular  forms 
of  categorical  proposition  on  the  one  hand,  and  what 
was  called  in  Part  I  Chapter  1 1 1  the  necessary  and  pos- 
sible forms  of  the  compound  proposition  on  the  other 
hand.  Thus,  the  form  'p  implies  q,'  where  the  relation 
asserted  of  the  two  component  propositions  is  irrespec- 
tive of  their  truth  or  falsity,  is  naturally  contradicted  in 
the  form  */  does  not  imply  q'  where  again  the  relation 
is  asserted  irrespectively  of  the  truth  or  falsity  of  these 
components.  Analogously,  taking/  and  q  to  be  adjec- 
tives (instead  of  propositions)  the  categorical  *  Every- 
thing that  is  /  is  ql  where  a  relation  oi  p  to  q  is 
asserted  irrespectively  oi^.ny given  thing  being/  or  q,  is 
naturally  contradicted  in  the  form  'Not  everything  that 
is  /  is-  q'  where  again  a  relation  is  asserted  of  /  to  ^ 
irrespectively  of  diny given  thing  being/  or  q.  Express- 
ing the  compound  propositions  in  terms  of  possible  or 
impossible,  the  proposition  '/  with  not-^  is  impossible,* 
contradicts  'p  with  not-^  is  possible,*  these  compounds 
being  respectively  analogous  to  the  universal  'Nothing 
that  is/  is  non-^'  and  the  particular  'Something  that  is 
/  is  non-^.'  Thus  there  is  a  literal  equivalence  in  the 
relations  subsisting  amongst  the  'necessary  composites' 
and  'possible  conjunctives'  on  the  one  hand,  and  those 
subsisting  amongst  the  'universal'  and  'particular'  cate- 


/ 


144 


CHAPTER  IX 


goricals  on  the  other  hand.  Some  logicians  indeed  have 
demanded  that  Logic  should  interpret  the  universal  and 
particular  categoricals  to  stand  respectively  for  neces- 
f  sary  implication  and  possible  conjunction ;  this  view, 
i  however  cannot  be  accepted.  The  analogy  that  properly 
holds  demands  equivalence — not  in  the  forms  of  pro- 
position themselves — but  in  the  logical  relations  amongst 
them.  The  basis  for  these  analogies  is  shown  in  the 
following  fundamental  forms,  where/  and  q  are  to  stand 
for  adjectives : 


General 
Categorical  Form 

A.    Every^  is  ^  ^  P 

A'.  Every  q\s  p  =  / 

£".    No/  is  ^  —  p 

E.  Everything  is  /  or  ^      ~  P 

0.  Not  every  p  is  ^  =  P 
(y.  Not  every  q  '\^  p 

1.  Some/  is  q 


Adjectivally 
Compound  Form 

implies 

is  implied  by 

is  co-disjunct  to 

is  co-alternate  to 

does  not  imply 


q 

q 
q 


S5  /    is  not  implied  by 
=  /  is  not  co-disjunct  to  q 
/'.    Not  everything  is/  or  ^  =s  /is  not  co-alternate  to  q 

The  four  relations  and  their  contradictories  here 
exhibited  lead  by  combination  to  seven  possible  rela- 
tions corresponding  exactly  to  those  shown  in  Part  I 
Chapter  III.    Thus: 


A2indA'.  "Every  p  is  q  and  Every  q  isp 

A  and  (7.  Every/  is  q  and  Not  every  q  isp 

^'and  O.  Every  q  isp  and  Not  every/  is  q 

O  and  (7  and  /  and  /'. 

E' SLndl.  Everything  isp  or  q  and  Some/  is  q 

E  and  /'.  No/  is  q  and  Not  everything  is/  or  q 

E2CcAE'.  No/  is  q  and  Everything  is/  or  q 


p  is  co-implicant  to  q 
p  is  super-implicant  to  q 
p  is  sub-implicant  to  q 
p  is  independent  of  q 
p  is  sub-opponent  to  q 
p  is  super-opponent  to  q 
p  is  co-opponent  to    q 


Thus  the  same  seven  relations  in  which  propositions 
may  stand  to  one  another  hold  of  the  relations  in  which 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    145 

adjectives  may  stand  when  entering  into  the  subject 
and  predicate  of  universal  and  particular  propositions. 

If/  is  formally  independent  of  ^,  then  all  the  rela- 
tions tabulated  above  are  material,  but  if  p  is  formally 
related  to  q,  then  it  must  be  related  in  one  or  other  of  the 
six  ways,  which  remain  when  the  relation  of  independ- 
ence is  excluded.  For  example,  taking  the  five  adjec- 
tives useful,  harmful,  useless,  harmless,  and  expedient, 
which  are  formally  related,  and  adding  pleasant  which 
is  formally  unrelated,  we  find  that: 

(1)  expedient  is  equipollent  to  useful, 

(ii)  useful  is  super-altern  to  harmless, 

(iii)  useless  is  sub-altern  to  harmful, 

(iv)  [useful  is  independent  of  pleasant], 

(v)  useless  is  sub-contrary  to  harmless, 

(vi)  useful  is  contrary  to  harmful, 

(vii)  useful  is  contradictory  to  useless. 

The  same  relations  hold  for  the  five  numerical  adjec- 
tives— greater  than  7,  less  than  4,  less  than  8,  less  than 
3,  not  less  than  8,  to  which  we  may  add  *even'  in  place 
of  *  pleasant.'    Thus: 


«b 


v>^" 


A 


^       ^^^^         contrary 


\ 


"^     sub-  contrary     ^ 


V     sub-  contrary    ^>S* 


o 
o. 
n 
•0 
n 
s 
a, 

3 


pleasant 


n 


even 


/ 


J.  L. 


ID 


146 


CHAPTER  IX 


§  7 .  Yet  another  way  of  representing  the  categorical 
proposition  is  in  terms— not  of  the  adjectives/  and  q— 
but  of  the  substantive  classes  P  and  Q  which  these 
adjectives  determine.  Any  class  must  be  conceived  in 
extension  as  a  part  of  the  universe  of  substantives,  where 
the  universe  is  sufficiently  widely  extended  to  include 
all  the  classes  which  occur  in  any  set  of  interconnected 
propositions.  The  absolutely  widest  substantive  uni- 
verse is  that  which  we  have  represented  by  the  word 
*  thing';  and  corresponding  to  any  more  restricted  uni- 
verse, the  same  word  'thing'  can  be  used  with  a  corre- 
spondingly understood  restriction. 

The  following  three  technical  terms  may  now  be 
introduced :  {a)  the  part  of  the  universe  which  remains 
when  any  given  class  is  subtracted  will  be  denominated 
the  remainder  (or  co-remainder)  to  the  class;  [b)  the 
smallest  class  that  includes  both  of  two  given  classes 
P  and  Q  will  be  denominated  'P  with  Q'\  and  {c)  the 
largest  class  that  is  included  in  both  of  two  given  classes 
P  and  Q  will  be  denominated  'P  into  Q\'  With  the 
help  of  these  three  class  functions,  the  following  funda- 
mental relations  between  the  class  functions  and  the 
adjectival  functions  which  determine  them  may  be 
expressed : 

(i)    The  class  determined  by  the  negative  not-/ =  the 
remainder  class  to  P, 

(2)  The  class  determined  by  the  adjectival  alternation 
*/  or  ^'  =  the  class  'P  with  Q,' 

(3)  The  class  determined  by  the  adjectival  conjunction 
'p  and  ^'  =  the  class  'P  into  Q' 

I        1  These  two  functions  of  P  and  Q  have  many  of  the  properties  of 
I  the  arithmetical  l.c.m.  and  h.c.f.    See  also  Chapter  VIII. 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    147 

Thus  the  notions  *not,'  *or,'  'and'  which  must  be  ap- 
plied to  predications  (and  here  to  adjectives)  correspond 
respectively  to  the  notions  'remainder/  'with,'  'into,' 
which  apply  to  substantive  classes.    Traditional  logic 
has  encouraged  confusion  between  these  two  types  of 
notion  by  employing  the  terms  which  are  only  proper 
for  adje:ctival  functions,  for  class  functions  also'.    This 
usage,  while  it  has  the  advantage  of  brevity  and  facili- 
tates the  logical  transformations  that  the  learner  has  to 
carry  out,  suffers  from  the  serious  objection  of  leading 
to  confusion  between  the  two  types  of  function.     Thus 
the  notion  of  the  remainder  as  a  relation  between  classes 
is  founded  upon  that  of  non-identity  as  a  relation  be- 
tween substantive  individuals.   For  when  the  class  X  is 
said  to  be  the  remainder  to  the  class  F,  part  of  what 
is  meant  is  that  no  individual  comprised  in  the  one  class 
is  identical  with  any  individual  comprised  in  the  other. 
Now  this  relation  of  non-identity  has  been  repeatedly! 
confounded  with  that  of  negation ;  so  much  so,  that  an  I 
important  school    of  philosophy  seems    to   hold    that 
diversity  or  non-identity  \nMo\ves  prima  facie  a  contra- 
diction; in  other  words,  that  the  togetherness  of  non- 
identical  substantives  in  the  universe  of  reality  involves 
the  joint  affirmation  and  negation  of  one  and  the  same 

predicate. 

The  correspondences  between  adjectival  relation- 
ships and  class  relationships  will  now  be  shown  by 
first  taking  each  of  the  four  universals  A,  A\  E,  E' , 
Thus: 


1  The  reversed  confusion  is  committed  when  adjectival  predications  | 
are  spoken  of  as  *  co-exclusive'  or  *  co-exhaustive'  instead  of  *co-  j 
disjunct'  or  *  co-alternate.' 


10 — 2 


148 


CHAPTER  IX 


Adjectival 
Relations 


Direct  Implicative 
Counter  Implicative 
Disjunctive 

Alternative 


Distributive  Relations 


p  implies  q 

p  is  implied  by  q 

p  and  q  are  co- 
disjunct 

^  and  $7  are  co- 
alternate 


A,  Every/  is  q 
A'.  Every  q'xsp 
E.    Nop  is  q 

E',  Everything  is/ or  $^ 


ass  Relations 


/included-in  2 

/^includes       ()\ 

Pd  Q  are  co 
jlusive 

P\d  Q  are  co- 
laustive 


Smce  O,  a,  I,  r  respectively  contradi.  A,  A', 
£,  E',  we  derive  the  following  combinatory  isults: 


(i)   /  is  co-implicant 
to  q 

(ii)  /  is  super-impli- 
cant  to  q 

(iii)/    is    sub-impli- 
cant  to  q 

(iv)  /  is  independent 
of  g 

(v)  /    is    sub-oppo- 
nent to  q 

(vi)  /  is  super-oppo- 
nent to  q 

(vii)  /  is  co-opponent 
to  q 


P  isco-incident   to 
P  issub- incident   to 


{A  and  A',)  Every  p  \^  q  and  Every 

g'^^P 

{A  and  O'.)  Every  p  \s  q  and  Not 
every  q\sp 

(^'and  O.)  Every  q  \s,  p  and  Not  P  is  ,uper-incident  to  I 
every/  is  ^  ' 

{OzxidO'\ 

and 
/and/'.)J 


P  is  inter-sectant   to 


Some  but  not  every  /  is 
q  and  Some  but  not 
every  non-/  is  non-q     \ 

{E'  and  /. )  Everything  is  either/  or    Z'  is  super-remainder  t 
q  and  Something   is 
both/  and  q 

{E  and  /'.)  Nothing  is  both/  and  q 
and  Not  everything  is 
/  or  ^ 

{E^ndE\)  Nothing  is  both  /  and  ^ 
and  Everything  is 
either/  or  q 


P  is  sub-remainder  tc! 


P  is  co-remainder  to 


Here  the  class-relationships  must  be  compared  and 
contrasted  with  the  adjective-relationships.  In  parti- 
cular  a  'super '-relation  for  the  adjectives  always  yields 
a  'sub '-relation  for  the  classes— illustrating  the  general 
pnnciple  that  a  more  determinate  connotation  yields  a 
I  narrower  denotation,  and  a  less  determinate  connotation 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    149 

yields  a  wider  denotation.    The  above  seven  relations 
may  be  at  once  expressed  on  Euler's  scheme.    Thus : 


(i) 

p  co-implicant  to   q 


(ii)  (iii) 

p  super-implJcant  to  j        p  sub-implicant  to  q 


P  co-incident  to   Q 


(iv) 
p   independent  of  q 


P  sub-incident  to  Q         P  super-incident  to  Q 
p  super-opponent  to  ^        p  sub-opponent  to  q 


p  co-opponent  to  q 


P  intcr-scctant  to  Q 


P  sub-remainder  to  Q 
(vi) 


P  super-remainder  to  Q 
(V) 


P  co-remainder  to  Q 
(vii) 

These  diagrams,  due  to  Euler,  illustrate  the  adapta- 
tion of  diagrammatic  representation  to  propositions  ex- 
pressed, as  above,  in  terms  of  classes.  The  employment 
of  diagrams  in  Logic  requires  some  special  explanation. 
A  class  is  represented  by  a  closed  figure,  while  any- 
thing that  is  comprised  in  the  class  may  be  represented 
by  a  point  within  this  figure ;  and  anything  not  com- 
prised in  the  class,  by  a  point  outside  the  figure.  It 
is  further  requisite  that  the  all-inclusive  class  (otherwise 
called  the  universe)  whether  restricted  or  unrestricted 
should  be  represented  also  by  a  closed  figure  within 
which  all  the  specific  classes  adjectivally  delimited  should 
fall.  Thus  the  boundary  line  may  be  taken  to  represent 
theadjectiveby  which  the  scope  of  the  class  is  determined, 


150 


CHAPTER  IX 


while  the  area  within  this  boundary-line  represents  the 
class  itself.  In  the  figure  below  the  class/*  is  represented 
as  determined  by  the  adjective/;  the  universe  is  repre- 
sented by  the  square,  and  what  is  outside  the  circle 
represents  the  class-remainder  to  Py  which  will  be  sym- 
bolized as  P\  the  boundary  of  which  is  indicated  by/. 
For  two  adjectives  p  and  q,  we  must  use  two  areas 
having  a  part  in  common  with  a  remainder  to  both. 
The  thickened  outline  in  (3)  separates  the  class  */^with 
Q'  from  'P'  into  Q^' \  and  the  thickened  line  in  (4) 
separates  the  class  'P  into  Q'  from  ' P'  with  Q\'  The 
diagram  shows  to  the  eye  the  correspondences  between 


the  adjectival  and  the  class  functions ;  viz.,  that  the  class 
determined  by  the  adjectival  alternation  'p  or  q'  is  the 
class  ^  P  with  Q',  and  that  determined  by  the  adjectival 
conjunction  'p  and  q'  is  the  class  ' P  into  Q'  These 
diagrams,  first  employed  by  Dr  Venn,  do  not  represent 
any  proposition,  but  the  framework  into  which  proposi- 
tions may  be  fitted.  Thus  it  is  shown,  for  instance,  that, 
using  two  determining  adjectives — p  and  q — the  uni- 
verse is  divided  into  2x2  classes,  namely :  P  into  Q^ 
P  into  Q ,  P'  into  Q,  and  P  into  Q\  determined  respec- 
tively by  the  adjectival  conjunctions  */  and^,'  'p  and  q^ 
*p  and  q,'  'p  and  q'  Again:  taking  three  determining 
adjectives  /,  q,  r,  we  must  draw  three  closed  figures 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    151 

in  such  a  way  that  every  resulting  sub-class  shall  be 

represented;  namely  the  2  x  2  x  2  classes 

P  into  Q  into  R,  P  into  Q  into  R\  F 

into  Q  into  R,  P'  into  Q  into  R,  P  into 

Q  into  i?,  P  into  Q  into  R\  P'  into  Q' 

into  R,  P'  into  Q  into  R\  as  determined 

by  the  corresponding  adjectival  conjunctions. 

Into  these  frameworks  propositions  are  fitted  in  the 
following  manner.  Beginning  with  a  single  determining 
adjective/,  consider  the  four  propositions:  A,  Every- 
thing is/;  E,  Nothing  is/;  /.  Something  is/;  O,  Not 
everything  is  /.  These  four  propositions  can  be  ex- 
pressed in  terms  of  the  classes  P  and  P'  thus:  A,  P 
exhausts  the  universe  or  P'  is  empty ;  E.  P  is  empty  or 
P^  exhausts  the  universe ;  /.  P  is  occupied  or  P'  does 
not  exhaust  the  universe ;  O.  P  does  not  exhaust  the 
universe  or  P'  is  occupied.  The  import  of  each  of  these 
propositions  may  therefore  be  expressed  by  means  of 
the  opposite  conceptions  of  occupied  and  empty :  crowded 
horizontal  shading  will  be  used  to  indicate  empty,  and 
a  single  straight  line  to  indicate  occupied : 


E 


Everything  is/ 


^Nothing  \&p 


Something  is  > 


Not  everything  is> 


This  shows  that  the  universals  may  be  expressed 
as  denying  and  the  particulars  as  affirming  occupation. 
Taking  now  two  adjectives/  and  q,  we  have  eight  dis- 
tinct propositions  A,  A\  E,  E\  O,  0\  /,  /',  where 


152 


CHAPTER  IX 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    153 


again  the  universals  deny  and  the  particulars  affirm  the 
occupation  of  certain  sub-classes. 


P  is  included  in  Q 


P  includes  Q 

r7T\ 

(  ^  0-   ) 

v^^^ 

P  and  Q  are  co-exclusive  P  and  Q  arc  co-cxhausuve 


An     Every  p  vs  q 


An     Every  q\s  p 


En       No  p  is  9 


£^  Everything  is  p  or  f 


These  are  respectively  contradicted  by 


Of     Not  all  p  is  ?  0{    Not  all  ^  is  p 


If  Something  is  p  and  q       I{     Not  everything 

is  p  or  9 


In  this  system  the  strict  relation  of  contradiction  is 
indicated  as  of  O  to  ^,  O'  to  A\  I  to  E,  /'  to  E\  by 
the  single  straight  line  and  the  shading  occurring  in  the 
same  sub-class,  namely  PQ\  P'Q,  PQ,  P'Q', 

It  must  be  observed  that  in  Venn's  system  (i)  the 
circles  are  drawn  in  every  case  as  overlapping  one 
another  without  exhausting  the  universe,  and  (2)  that 
the  specific  proposition  is  represented  by  marking  some 
sub-class  as  occupied  or  as  empty. 

Next  let  us  combine  the  diagrams  representing  the 
several  universals,  which  contain  no  instantial  affirma- 
tion, with  the  diagram  representing  'Something  is  /' 
for  A  and  E,  and  'Something  is  not-/'  for  A'  and  E'\ 
or  again  with  the  diagram  representing  'Something  is  q' 
for  A  and  E\  and  'Something  is  q  for  A'  and  E, 


A  Nothing  is  pq 
and  something  is  p 


E  Nothing  is  P9 
and  something  is  p 


A   Nothmg  is  pq 
and  something  is  f 


E'  Nothmg  IS 

w 

and  something 

is  ^ 

rA 

Ih^ 

A  Nothing  is  pq 
and  something  is  q 


E  Nothing  is  pq 
and  something  is  q 


A'  Nothing  is  "pq 
and  something  is  q 


E'  Nothing  is  fq 
and  something  is  q 


Now  since  a  single  line  means  that  something  is 
to  be  found  in  one  or  other  of  the  two  sub-classes  which 
it  crosses,  and  since  the  shading  denies  it  for  one  of  the 
two,  it  follows  that  something  is  to  be  found  in  the 
other.  Thus  the  first  four  diagrams  prove  to  the  eye 
respectively  that: 

From  'Every/ is  q  and     Something  is     /'  we  can  infer/.    'Some/  is  q' 
From  ^  No     /  is  ^  and    Something  is    /'we  can  infer  O.  '  Some  /  is  non-^' 
From  'Every  q  is/  and  Something  is  non-/'  we  can  infer  /'.  'Some  non-^  is  non-q 

From  '  Every-      1  ^^^^  Something  is  non-/ '  we  can  infer  O,  *  Some  non-/  is  q ' 
thmg  is/  or  qj 

And  similarly  for  the  second  four,  where  the  affirmative 
instantial  involves  non-^  or  q. 

These  inferences  illustrate  the  general  principle  that 
in  order  to  infer  a  proposition  giving  instantial  affirma- 
tion, we  must  have  a  premiss  giving  instantial  affirma- 
tion. From  A„  alone  we  cannot  infer  /^,  but  from  A^ 
jointly  with  'Something  is/'  we  can  infer  If, 


154 


CHAPTER  IX 


Finally  let  us  use  Venn's  diagrams  to  represent  the 
seven  relations  which  result  from  the  possible  com- 
binations of  the  eight  elementary  propositions  A^,  AJ, 


•^ni    -^^ 


n  f    -'/•»  -^/f    ^fi    ^f' 


(i) 


yC           ■%:  "  jT 

^ 

V'      X 

1 

(ii) 


(iii) 


/4  and  /4'  and  /  and  /' 


»5 


0 

i 

b 

/4  and  Cr  and  /  and  /'   /I' and  O  and  /  and  /' 


(vi) 


(vii) 

D: 

0 

ffl 

D 

/  and  /'  and  O  and  Cf 


E  and  /'  and  O  and  O'  £'  and  /  and  O  and  (7 


£  andJ?'  and  O  and  CX 


In  comparing  this  scheme  with  that  of  Euler  (pre- 
viously given)  two  points  arise.  Euler  draws  these  figures 
from  the  outset  so  as  to  represent  the  class-relationships 
both  as  regards  instantial  affirmations  and  instantial 
denials,  so  that  the  figures  directly  express  propositional 
information.  But  in  Venn  both  these  factors  in  the 
proposition  have  to  be  specifically  marked  and  in  order 
to  represent  a  completely  determined  class-relationship 
all  the  four  sub-classes  must  be  marked.  In  spite  of  this 
apparent  difference,  an  optical  comparison  of  this  last 
scheme  with  the  Eulerian  scheme  on  p.  149  will  disclose 
their  essential  equivalence.  The  practical  distinction, 
however,  remains  that  in  Euler  s  scheme  each  uncom- 


GENERAL  PROPOSITION  AND  ITS  IMPLICATIONS    155 


pounded  categorical  must  be  represented  by  an  a/^er- 

native  of  figures,  viz. : 

A  by  (i)  or  (ii) ;  and  0  by  (iii)  or  (iv)  or  (v)  or  (vi)  or  (vii) 

A  by  (i)or  (iii) ;  and  O'  by  (ii)  or  (iv)  or  (v)  or  (vi)  or  (vii) 

E  by  (vi)  or  (vii) ;  and  /  by  (i)  or  (ii)  or  (iii)  or  (iv)  or  (v) 

E'  by  (v)  or  (vii) ;  and  /'  by  (i)  or  (ii)  or  (iii)  or  (iv)  or  (vi) 

and  conversely, 

each  diagram  represents  a  conjunction  of  propositions 

(i)   =A  and  A\  (ii)  =  ^  and  O' ,  (iii)=^'  and  O, 
(iv)  =/ and /'and  O  and  O' , 

(v)  ^E'  and  /,  (vi)  =^  and  I\  (vii)  =  ^  and  E\ 

On  the  other  hand  Venn's  diagrams  represent  each  of 
the  uncompounded  propositions  by  its  appropriate  *  mark- 
ing' of  the  proper  sub-class,  and  are  thus  immediately 
adapted  to  the  conjunction  of  two  or  more  affirmatively 
or  negatively  instantial  pieces  of  information. 

§  8.  All  the  above  inferential  operations  are  per- 
formed upon  adjectival  factors,  these  occurring  always 
as  predicates  in  a  principal  or  subordinate  clause;  and, 
as  is  impressively  brought  out  in  the  so-called  '  exist- 
ential' formulation  of  the  proposition,  a  residual 
substantival  factor  always  remains  in  the  subject,  though 
for  linguistic  convenience  it  may  appear  also  in  the 
predicate.  The  importance  of  this  feature  may  have 
been  obscured  owing  to  the  complicated  detail  with 
which  the  inferences  have  been  treated  ;  and,  in  con- 
clusion, it  is  therefore  to  the  point  to  emphasize  the 
connection  between  the  account  of  inference  in  this 
chapter  and  that  of  the  functioning  of  substantive  and 
adjective  given  in  Chapter  I. 


156 


CHAPTER  X 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE 

PROPOSITIONS 

§  I.  Before  directly  approaching  the  topic  to  be 
dealt  with  in  this  chapter,  it  will  be  necessary  to  con- 
sider the  general  question  of  the  classification  of  pro- 
positions. In  previous  chapters,  several  classifications 
of  propositions  under  different  fundanienta  divisionis 
have  been  given :  for  example  they  have  been  divided 

a.  into  simple  and  compound,  the  latter  being  subdivided 
into  conjunctive  and  composite;  and  again  into  uncer- 

^   tified  and  certified,  the    latter  being  subdivided   into 
formally  and  experientially  certified;  even  the  distinc- 

f.  tion  between  true  and  false  yields  an  exclusive  and 
exhaustive  division  of  propositions.  On  the  other  hand, 
many  well-known  so-called  classifications  of  proposi- 
tions break  the  purely  formal  rules  of  logical  division, 
in  that  the  sub-classes  are  not  mutually  exclusive,  and 
often  can  hardly  be  regarded  as  collectively  exhaustive. 
The  most  notorious  example  of  this  is  the  classification 
of  propositions  upon  which  Kant  based  his  enumeration 
of  the  categories,  and  which  comprised  such  sub-classes 
as  singular,  particular,  universal,  affirmative,  negative, 
^hypothetical,  categorical,  assertoric  and  problematic. 
Regarded  as  a  classification  of  propositions  this  involves 
a  flagrant  violation  of  the  formal  rules  of  division ;  for 
a  categorical  proposition  may  be  singular  or  universal, 
negative  or  affirmative,  problematic  or  assertoric,  etc. 
What  is  of  real  logical  value,  and  was  indeed  intended 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    157 


by  Kant,  is  a  classification  not  of  propositions,  but  of 
the  several  formal  relations  which  may  enter  within  the 
structure  of  a  proposition  more  or  less  simple  or  com- 
■  plex.  For  example,  Tf  the  American  harvest  is  bad, 
the  European  prices  of  corn  are  high'  is  properly 
enough  denominated  hypothetical  because  the  central 
or  principal  relation  asserted  is  that  of  implication ;  but 
further  analysis  discloses  the  categorical  nature  of  the 
two  implicationally  related  clauses,  and  the  universality 
of  the  statement  as  understood  to  refer  to  any  or  every 
year;  and  furthermore  such  relations  as  contemporaneity 
and  causality  may  be  taken  as  implicitly  asserted  in  such 
a  proposition.  Of  distinctions  amongst  the  different 
forms  of  relation  that  may  enter  into  a  proposition  we 
may  select  as  one  of  the  most  important  that  between 
the  relation  of  characterisation  and  the  relation  of  im- 
plication, which,  properly  speaking,  should  take  the 
place  of  the  distinction  amongst  propositions  between 
categorical  and  hypothetical.  Of  these  two  relations — 
characterisation  and  implication — the  former  holds  only 
of  an  adjective  to  a  substantive,  the  latter  only  of  one 
proposition  to  another  proposition.  Again,  the  distinc- 
tions of  modality  are  not,  properly  speaking,  distinctions 
between  propositions,  but  distinctions  between  the  dif- 
ferent adjectives  that  can  be  significantly  predicated  of 
propositions.  In  short  the  sole  logical  purport  of  a 
so-called  classification  of  propositions  is,  by  means  of 
an  analysis  of  propositions  of  various  forms  of  com- 
plexity, to  disclose  the  different  modes  in  which  their 
components  are  bound  into  a  unity. 

With  special  reference  to  the  topic  of  this  chapter 
we  may  pass  to  such  logicians  as  Lotze,  Bosanquet 


158 


CHAPTER  X 


and  others  who  have  attempted  to  classify  propositions 
on  philosophical  rather  than  purely  formal  principles. 
In  particular  Sigwart  has  distinguished  propositions 
under  three — not  necessarily  exclusive  or  exhaustive — 
heads  corresponding  to  what,  in  the  title,  we  have  called 
existential,  subsistential  and  narrative. 

§  2.  We  proceed  then  to  examine  in  the  first  place 
what  is  meant  by  an  existential  proposition.  The  most 
general  and  appropriate  sense  in  which  the  word  'exist- 
ential* is  predicated  of  a  proposition  is  where  the  pro- 
position refers  to  existence  in  that  narrower  sense  in 
which  existence  is  distinguished  from  subsistence,  as  two 
sub-divisions  of  reality.  Thus  the  proposition  '3  plus 
4  equals  7'  must  be  regarded  not  as  existential,  but  as 
subsistential,  if  that  terminology  be  permitted.  From 
an  examination  of  the  illustrations  given  by  philosophers 
^  of  the  existent  (as  distinct  from  the  subsistent)  it  may 
be  gathered  that  the  term  is  equivalent  to  that  which  is 
manifested  in  time  or  space.  This  interpretation  may  be 
justified  by  considering  the  etymology  of  the  word 
*  existent'  which  is  closely  connected  with  *  external,' 
and  is  further  confirmed  by  the  fact  that  the  typical 
so-called  external  relations  are  temporal  or  spatial.  On 
the  other  hand  it  has  been  maintained,  for  example, 
that  the  number  3,  or  the  relation  of  equality  between 
3  plus  4  and  7,  or  the  relation  of  causality  subsists 
rather  than  exists.  If  this  conception  is  to  be  general- 
ised what  subsists  is  primarily  an  adjective,  whether 
•  ordinary  or  relational;  whereas  what  in  the  more  exact 
'  sense  may  be  said  to  exist  is  a  substantive  proper.  We 
may  therefore  regard  the  terms  *  existent'  or  'substan- 
tive proper'  as  meaning  'what  is  manifested  in  time  or 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    159 

space.'  Thus  an  existential  proposition  is  distinguished 
from  a  subsistential  proposition  in  that  the  latter  makes 
predications  about  adjectives  (including  propositions) 
as  such.  It  may,  however,  be  maintained  that  such  a 
proposition  as  '3  plus  4  equals  f  should  properly  be 
interpreted  as  existential  on  the  ground  that  it  applies 
to  all  possible  existent  groupings  of  classes  numbered 
3,  4,  and  7;  or  again  that  the  proposition  'Unpunctu- 
ality  is  irritating' is  existential  on  the  ground  that  it 
means  nothing  more  nor  less  than  that  'All  unpunctual 
arrivals  are  irritating,'  where  the  term  arrival,  with  its 
implicit  temporal  and  spatial  reference,  obviously  stands 
for  an  existent.  Or  yet  again  that  the  proposition  '  Heat 
causes  wax  to  melt'  is  existential  on  the  ground  that  it 
merely  expresses  the  universal  proposition  that  'AH 
cases  in  which  heat  enters  wax,  are  cases  in  which  the 
wax  is  melted,'  where  merely  temporal  and  spatial  rela- 
tions of  possible  occurrences  are  involved.  If  then  a 
subsistential  proposition  is  to  be  distinguished  from  an 
existential,  it  must  be  on  the  ground  that  propositions 
in  which  the  explicit  predications  concern  adjectives  or 
relations,  have  a  special  significance  beyond  what  they 
undoubtedly  imply  existentially^ 

§  3.  But  turning  from  the  more  philosophical  to 
the  strictly  formal  usage  of  the  term  existential,  we  find 
that  by  such  logicians  as  Venn,  Keynes  and  Russell, 
existential  and  subsistential  propositions  are  indiffer- 
ently denominated  existential,  and  that  the  term  exist- 
ential is  used  without  any  reference  to  the  substance  of 
the  proposition,  but  rather  to  a  certain  mode  in  which 

*  This  question  will  be  treated  in  more  detail  in  a  subsequent 
chapter. 


i6o 


CHAPTER  X 


ill 


^.ny general  proposition  (particular  or  universal)  may  be 
formulated.  This  entirely  distinct  and  peculiar  use  of 
the  term  'existential'  has  given  rise  to  endless  confu- 
sion ;  and,  on  this  account,  the  term  should  be  entirely 
discarded  and  replaced  by  some  such  term  as  instantialy 
or,  more  accurately,  indeterminately  instantial.  At  this 
point  we  must  explain  the  distinction  between  determin- 
ately  instantial  and  indeterminately  instantial.  While  the 
former  corresponds  roughly  to  narrative  propositions, 
of  which  we  shall  treat  later,  the  latter  are  most  natur- 
ally prefixed  by  the  phrase  'there  is'  or  'there  are'; 
(e.g.  there  is  a  God;  there  are  horses;  there  are  no 
sea-serpents ;  there  is  an  integer  between  3  and  5 ;  there 
are  prime  integers  between  4  and  15;  there  is  no  in- 
teger between  3  and  4.  Of  these,  the  first  three  would 
be  called  existential,  in  the  philosophical  sense,  the  last 
three  subsistential.  A  minor  distinction  amongst  such 
indeterminately  instantial  propositions,  the  disregard 
of  which  has  not  infrequently  led  to  confusion,  is  that 
between  the  affirmatively  instantial  and  the  negatively 
instantial.  In  short,  the  essential  nature  of  a  particular 
or  of  a  universal  proposition  is  expressed  by  formulating 
the  former  as  affirmatively  instantial,  and  the  latter  as 
negatively  instantiaP. 

The  further  development  of  the  topic  of  such  pro- 
positions, or  rather  of  such  propositional  formulations, 
requires  us  to  introduce  the  phrase  'universe  of  dis- 
course,' to  which  frequent  reference  is  made  in  formal 
expositions  of  the  so-called  existential  import  of  pro- 
positions. There  are  two  applications  of  this  phrase, 
which  demand  different  criticisms.    One  quite  harmless 

^  See  Chapter  VIII. 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    161 

application  of  the  expression  'universe  of  discourse,' 
points  merely  to  the  familiarly  elliptical  nature  of  con-  (/> 
versation.  Thus  the  reference  of  such  a  proposition  as 
'All  voters  are  males'  is  understood  to  be  limited,  say, 
to  the  present  time  (1914),  the  English  nation,  and 
election  to  Parliament.  In  spite  of  the  fact  that  some 
nations  now  and  all  nations  will  confer  the  franchise 
upon  women,  in  spite  of  the  fact  that  for  the  Board  of 
Guardians  and  other  offices  women  take  part  in  voting, 
the  proposition  'all  voters  are  males'  is  perfectly  intel- 
ligible in  its  context.  The  phrase,  though  elliptical  — 
like  all  phrases  in  discussion  or  conversation — does  not 
require  the  explicit  introduction  of  every  well-under- 
stood qualification.  I  n  our  view,  therefore,  logicians  have 
unnecessarily  paraded  this  application  of  the  notion  of 
a  universe  of  discourse,  where  it  means  merely  that  the 
ordinary  reader  is  expected  to  supply  the  restrictions 
intended  by  the  writer.  A  limited  universe  understood 
as  indicating  the  subject-matter  of  a  single  work,  such 
as  geometry,  which  refers  exclusively  to  spatial  figures, 
illustrates  the  same  simple  relation  to  the  universe  as  a 
whole.  Understood  in  this  general  sense,  the  universe 
of  discourse  has  to  the  universe  the  relation  of  part  to 
whole,  and  the  notion  is  certainly  harmless  if  trivial. 

But  the  other  application  of  the  phrase  requires 
more  serious  criticism.  Here  'the  universe  of  discourse'  ^  -' 
is  presented  to  the  reader,  not  as  inside,  but  as  out- 
side what  is  commonly  called  the  universe.  In  this  usage 
the  phrase  seems  to  imply  that  there  are  several  uni- 
verses related  to  one  another  as  Europe  is  to  Africa 
rather  than  as  France  is  to  Europe,  taking  Europe  in 
both  cases  to  stand  for  the  universe.    The  distinction 


J.  L. 


II 


l62 


CHAPTER  X 


p«^. 


between  the  two  uses  of  the  phrase  is  evident  when  we 
pass  from  such  an  example  as  ^  All  voters  are  males'  to 
'Some  fairies  are  malevolent'  which  well  illustrates  the 
use  now  under  consideration.   The  former  is  understood 
to  refer  to  a  limited  part  of  the  universe  of  persons, 
whereas  the  latter  refers  to  no  part  whatever  of  this 
universe,  and  on  this  account  is  said  to  be  concerned 
with  the  *  universe  of  imagination'  conceived  as  outside 
and  separate  from  the  universe  of  reality.    Now  if,  when 
speaking  of  specific  universes,  such  as  the  universe  of 
imagination,  the  universe  of  ideation,  and  the  universe 
of  physical  reality,  we  meant  merely  universes  comprising 
images,  ideas,  or  physical  realities,  then  all  these  three 
are  in  the  strictest  sense  included  within  the  one  single 
universe  of  existents,  to  which  they  are  related  merely 
as  parts  to  whole.    Anything  that  is  comprised  in  the 
universe  of  images  must  be  an  image;  anything  that 
is  comprised  in  the  universe  of  ideas  must  be  an  idea 
(both  of  these  being  psychical) ;  and  similarly,  anything 
comprised  in  the  universe  of  physical  realities  must  be 
physical.    What  logicians  seem  to  have  confused,  and 
requires  only  common  sense  to  distinguish,  is  between 
a  horse  and  either  the  idea  of  a  horse  or  the  image  of 
a  horse ;  and  accordingly  a  proposition  about  horses  is 
.  concerned  with  different  material  from  any  proposition 
j  about  ideas  or  images  of  horses.    When  then  a  proposi- 
tion is  spoken  of  as  being  false  in  the  universe  of  reality 
and  yet  true  in  the  universe  of  imagination  or  ideation, 
this  involves  the  tacit  assertion  that  the  »ame  proposi- 
tion can  be  both  true  and  false;  whereas  in  fact  the 
contents  of  the  two  propositions,  one  of  which  is  said  to 
be  true  and  the  other  false, are  different.  The  affirmation 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    163 

or  denial  that  there  are  sea-serpents  is  different  from 
the  affirmation  or  denial  that  there  are  images  of  sea- 
serpents  ;  which  again  is  different  from  the  affirmation  or 
denial  that  there  are  ideas  of  sea-serpents.  It  is  absurd 
to  say  that  the  same  things  exist  in  one  universe  and  do 
not  exist  in  another:  wherever  this  appears  to  be  the 
case  the  things  asserted  or  denied  to  exist  are  different. 
What  is  here  said  of  sea-serpents  holds  equally  of  horses 
or  of  dragons ;  as  regards  the  latter,  it  is  supposed  that 
because  dragons  are  acknowledged  not  to  exist  in  the 
universe  of  physical  reality,  there  must  be  some  uni- 
verse in  which  they  do  exist  in  order  that  we  may 
intelligently  use  the  term  'dragon.'  Now  it  is  a  purely 
psychological  question  whether  at  this  or  that  moment 
of  time  an  image  of  a  horse  or  equally  of  a  dragon  is 
in  course  of  mental  construction;  it  may  be  that  we 
may  intelligently  read  or  think  about  dragons  or  horses 
without  mentally  constructing  any  images  of  such  crea- 
tures. Properly  speaking  there  is  no  such  thing  as  the 
image  of  a  horse  or  the  image  of  a  dragon,  because  the 
constructing  of  images  by  one  person  at  one  time  is, 
as  an  occurrence,  distinct  from  such  a  construction  by 
another  person  at  another  time,  however  closely  these 
images  may  agree  with  one  another  in  character.  Hence, 
if  existence  is  predicated  of  any  image  of  this  or  that 
kind,  it  must  be  remembered  that  by  existence  is  here 
meant  manifestation  in  time,  and  that  therefore  there 
exist  as  many  images  of  any  kind  of  thing  as  there  are 
occurrences  of  the  constructive  act. 

What  holds  of  images  holds,  strictly  speaking,  also 
of  ideas,  though  not  so  obviously ;  the  existence  or 
non-existence  of  the  idea  of  any  object,  if  idea  stands 

II— 2 


^ 


164 


CHAPTER  X 


for  mental  process,  must  mean  the  occurrence  or  non- 
occurrence of  an  act  of  thinking  about  the  object  during 
this  or  that  period  of  time.  The  term  *idea,'  however, 
may  be  understood  in  a  less  literally  psychological  sense : 
thus  intelligently  to  entertain  a  proposition  in  thought 
would  seem  to  entail  our  entertaining  ideas  correspond- 
ing to  the  several  terms  in  the  proposition.  But  in  this 
connection  we  may  refer  to  Mill's  pronouncement  in 
regard  to  the  import  of  propositions  in  relation  to  ideas. 
His  dictum  is  that  propositions  are  not  about  ideas,  but 
about  things;  and  by  this  he  intended  to  assert  that  a 
proposition  is  concerned  with  the  things  which  it  ex- 
pressly talks  of,  and  not  with  any  mental  process  that 
j  may  be  involved  in  the  assent  to  or  understanding  of 
Uhe  proposition.  In  short,  although  any  genuine  act  of 
assertion  requires  as  a  preliminary  process  the  under- 
standing of  the  terms  and  combination  of  terms  that 
constitute  a  proposition,  yet  it  is  not  this  process  to 
which  the  proposition  refers.  This,  of  course,  holds, 
whether  the  matter  of  the  proposition  is  physical  reality 
or  mental  reality:  we  must  understand  what  is  meant 
by  the  association  of  ideas  or  by  an  emotion  of  anger 
in  a  psychological  proposition,  just  as  we  must  under- 
stand what  is  meant  by  dragons  or  horses  in  proposi- 
tions describing  such  creatures;  while,  on  the  other 
hand,  the  propositions  in  neither  case  are  concerned 
with  these  processes  of  understanding. 

§  4.  We  next  proceed  to  consider  in  what  sense 
truth  and  falsity  can  be  predicated  of  propositions  such 
as  'Some  fairies  are  malevolent'  or  *No  Greek  gods 
are  without  human  frailties.'  These  may  be  otherwise 
rendered:  'There  are  malevolent  fairies,'  *  There  are  no 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    165 

Greek  gods  without  human  frailties.'     If  these  be  taken 
literally,  as  merely  primary  propositions,  nothing  can  be 
said  but  that  the  first  is  necessarily  false  and  the  second 
necessarily  true,  because  there  are  actually  no  fairies  and 
no  Greek  gods  in  the  real  universe.    But  within  the  real 
universe  there  are  to  be  found  descriptions  of  fairies 
and  of  Greek  gods  in  stories  or  legends,  and  hence  it 
may  be  true  that  some  fairies  have  been  described  as 
malevolent,  and  it  may  h^  false  that  no  Greek  gods  have 
been  described  as  without  human  frailties.     If  then  we 
distinguish  these  secondary  propositions — to  be  recog- 
nised as  such  by  the  introduction  of  the  word  'describe* — 
from  the  original  primary  propositions,  the  establish- 
ment of  their  truth  or  falsity  is  seen  to  depend  upon 
special  evidence.    The  universe  of  descriptions  is  simply 
part  of  the  universe  of  reality;  indeed  it  seems  strangely 
to  have  escaped  logicians  that  books  and  the  persons 
who  wrote  them  belong  to  one  real  world,  and  that 
therefore  the  universe  to  which  we  refer  for  verification 
of  propositions  concerning  the  descriptions  of  fairies  or 
of  Greek  gods  is  simply  and  precisely  the  same  universe 
as  that  to  which  we  refer  for  verification  of  propositions 
concerning  Frenchmen  and  geologists.    It  is  therefore 
evident  that,  only  when  we  have  transformed  such  pri- 
mary propositions  into  their  secondary  correspondents, 
any  question  of  interest  arises  as  to  their  truth  or  falsity. 
This  contention  finally  forbids  us  to  speak  of  various 
universes  of  discourse  which  are  outside  the  one  uni- 
verse of  reality.    The  briefest  mode  of  indicating  the 
peculiarity  of  propositions  of  the  type  illustrated  is  to  say 
quite  simply  that  they  are  elliptical ;  not  elliptical  in  the 
sense  of  limiting  the  subject-term  to  a  narrower  sphere 


i66 


CHAPTER  X 


included  in  the  universe,  but  elliptical  in  the  sense  of 
being  expressed  as  primary  propositions  and  understood 
as  secondary.  Thus  our  first  example  should  properly  be 
expressed  'Story-books  describe  some  fairies  as  being 
malevolent,'  and  our  second  ^  Homer  describes  all  the 
Greek  gods  as  subject  to  human  frailties';  and  in  these 
transformed  shapes  the  propositions  are  seen  at  once 
to  be  verifiable  in  exactly  the  same  way  as  any  other 
propositions ;  namely  by  reference  to  the  one  real  uni- 
verse of  books  and  persons. 

§  5-    We  pass  now  to  the  logical  significance  of  the 
term  narrative  in  its  application  to  propositions.     The 
notion  of  a  narrative  proposition  is  not  restricted  to  the 
type  of  proposition  characteristic  of  a  work  of  fiction  or 
history,  since  it  includes  statements  made  in  ordinary 
conversation  etc.,  where  there  may  be  no  intention  to 
develop  the  account  of  an  incident  into  a  connected 
story.    Moreover  histories  and  novels  are  composed  of 
others  besides  narrative  propositions — the  non-narrative 
propositions  being  generally  what  we  may  call  comments 
on  the  incidents,  characters,  situations  or  emotions  de- 
scribed.   Novels  (or  even  histories)  might  indeed  be 
classified  according  as  their  narrative  or  commentary 
elements  predominate;  compare  for  example  Scott  with 
Thackeray,  or  S.  R.  Gardiner  with  Macaulay.    A  nar- 
rative proposition  may  be  more  precisely  defined  as  one 
whose  subject-term  is  prefixed  by  introductory  or  refer- 
ential applicatives ;  whereas  non-narrative  propositions 
are  prefixed  by  such  distributives  as  'every,'  'some'  or 
similar  phrases.    Now  distributives  serve  as  predesigna- 
tions  of  adjectivally  significant  subjects,  while  com- 
mentary  propositions  may  be  distinguished  from  such 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    167 

narrative  propositions  as  may  happen  to  use  subjects  con- 
taining an  adjectival  element  because  in  the  latter  case 
the  adjective  has  no  general  reference.    For  example, 
in  the  course  of  a  narrative  the  proposition  may  occur 
*A  shabbily-dressed  gentleman  entered  the  room,'  and 
this  may  be  followed  later  on  by  'the  shabby  gentleman 
withdrew,'  where  the  adjective  'shabby'  enters  in  the 
context  merely  first  as  introductory  and  later  as  refer- 
ential.   In  fact  it  is  not  by  the  consideration  merely  of 
the  grammatical  structure  of  a  sentence,  but  rather  by 
the  logical  nexus  of  the  propositions  that  the  distinction 
can  be  established;   the  narrative   and   non-narrative 
elements  in  any  literary  work  being  not  necessarily  ex- 
pressed in  separable  sentences.    Hence  the  reader  may 
easily  pick  out  the  commentary  elements,  these  being 
recognisable  by  their  reference  to  persons  and  things  in 
gene'ral  as  distinct  from  the  persons  and  things  entering 

directly  into  the  plot. 

My  account  of  narrative  propositions  covers  a  wider 
range  than  is  apparently  intended  by  Sigwart;  but  for 
both  of  us  the  distinction  between  narrative  and  non- 
narrative  rests  upon  that  between  the  substantive  and 
the  adjective.  In  Sigwart's  definition  the  subject  in  the 
narrative  proposition  is  merely  substantival,  while  the 
subject  as  well  as  the  predicate  in  the  non-narrative 
proposition  contains  adjectival  elements.  My  applica- 
tion of  the  term  'narrative'  on  the  other  hand,  includes 
cases  in  which  the  subject  term  may  contain  an  adjectival 
element  the  significance  of  which  is  purely  introductory 

or  referential. 

§  6.    A  new  problem,  bearing  upon  the  existential 
import    of   propositions    is    raised  when  we  contrast 


i66 


CHAPTER  X 


included  in  the  universe,  but  elliptical  in  the  sense  of 
being  expressed  as  primary  propositions  and  understood 
as  secondary.  Thus  our  first  example  should  properly  be 
expressed  'Story-books  describe  some  fairies  as  being 
malevolent,' and  our  second  'Homer  describes  all  the 
Greek  gods  as  subject  to  human  frailties';  and  in  these 
transformed  shapes  the  propositions  are  seen  at  once 
to  be  verifiable  in  exactly  the  same  way  as  any  other 
propositions ;  namely  by  reference  to  the  one  real  uni- 
verse of  books  and  persons. 

§  5.    We  pass  now  to  the  logical  significance  of  the 
term  narrative  in  its  application  to  propositions.     The 
notion  of  a  narrative  proposition  is  not  restricted  to  the 
type  of  proposition  characteristic  of  a  work  of  fiction  or 
history,  since  it  includes  statements  made  in  ordinary 
conversation  etc.,  where  there  may  be  no  intention  to 
develop  the  account  of  an  incident  into  a  connected 
story.    Moreover  histories  and  novels  are  composed  of 
others  besides  narrative  propositions — the  non-narrative 
propositions  being  generally  what  we  may  call  comments 
on  the  incidents,  characters,  situations  or  emotions  de- 
scribed.   Novels  (or  even  histories)  might  indeed  be 
classified  according  as  their  narrative  or  commentary 
elements  predominate;  compare  for  example  Scott  with 
Thackeray,  or  S.  R.  Gardiner  with  Macaulay.    A  nar- 
rative proposition  may  be  more  precisely  defined  as  one 
whose  subject-term  is  prefixed  by  introductory  or  refer- 
ential applicatives ;  whereas  non-narrative  propositions 
are  prefixed  by  such  distributives  as  'evei^,'  'some'  or 
similar  phrases.    Now  distributives  serve  as  predesigna- 
tions  of  adjectivally  significant  subjects,   while  com- 
mentary propositions  may  be  distinguished  from  such 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    167 

narrative  propositions  as  may  happen  to  use  subjects  con- 
taining an  adjectival  element  because  in  the  latter  case 
the  adjective  has  no  general  reference.    For  example, 
in  the  course  of  a  narrative  the  proposition  may  occur 
'A  shabbily-dressed  gentleman  entered  the  room,'  and 
this  may  be  followed  later  on  by  'the  shabby  gentleman 
withdrew,'  where  the  adjective  'shabby'  enters  in  the 
context  merely  first  as  introductory  and  later  as  refer- 
ential.   In  fact  it  is  not  by  the  consideration  merely  of 
the  grammatical  structure  of  a  sentence,  but  rather  by 
the  logical  nexus  of  the  propositions  that  the  distinction 
can  be  established;    the  narrative   and    non-narrative 
elements  in  any  literary  work  being  not  necessarily  ex- 
pressed in  separable  sentences.    Hence  the  reader  may 
easily  pick  out  the  commentary  elements,  these  being 
recognisable  by  their  reference  to  persons  and  things  in 
general  as  distinct  from  the  persons  and  things  entering 

directly  into  the  plot. 

My  account  of  narrative  propositions  covers  a  wider 
range  than  is  apparently  intended  by  Sigwart;  but  for 
both  of  us  the  distinction  between  narrative  and  non- 
narrative  rests  upon  that  between  the  substantive  and 
the  adjective.  In  Sigwart's  definition  the  subject  in  the 
narrative  proposition  is  merely  substantival,  while  the 
subject  as  well  as  the  predicate  in  the  non-narrative 
proposition  contains  adjectival  elements.  My  applica- 
tion of  the  term  'narrative'  on  the  other  hand,  includes 
cases  in  which  the  subject  term  may  contain  an  adjectival 
element  the  significance  of  which  is  purely  introductory 

or  referential. 

§  6.    A  new  problem,  bearing  upon  the  existential 
import    of  propositions    is    raised  when  we  contrast 


i68 


CHAPTER  X 


\ 


fictitious  with  historical  narratives.  We  may  take  for 
illustration  the  proposition  'Mr  Pecksniff  is  a  hypocrite' 
and  first  ask  what  is  meant  by  Mr  Pecksniff.  Now  a 
provisional  answer  to  this  would  be  'A  certain  archi- 
tect, living  near  Salisbury,  in  the  beginning  of  the  nine- 
teenth century.'  The  term  by  which  we  have  replaced 
Mr  Pecksniff  seems  to  have  an  obvious  reference  to 
the  universe  of  reality,  and  more  particularly  to  the 
universe  of  things  happening  and  existing  in  time  and 
place.  But  the  question  as  to  whether  there  was  any 
architect  in  Salisbury  at  that  time  would  be  irrelevant, 
and  therefore  the  proposition  would  appear  not  to  be 
about  any  architect  then  living  near  Salisbury.  The 
difficulty  here  points  to  a  peculiarity  in  the  use  of  the 
predesignation  'a  certain.'  If  for  Pecksniff  we  had 
substituted  'some  architect'  instead  of  *a  certain  archi- 
tect,' the  proposition  *Some  architect  living  then  near 
Salisbury  was  a  hypocrite'  would  have  been  amenable 
to  the  ordinary  modes  of  verification.  But  the  form  of 
statement  *A  certain  architect  was  a  hypocrite'  appears 
not  to  represent  a  proposition,  inasmuch  as  it  cannot 
be  either  affirmed  or  denied,  since  the  architect  to 
whom  the  writer  refers  is  not  indicated.  What  holds 
then  of  the  reader  or  hearer  of  such  a  proposition  does 
not  hold  of  the  writer  or  speaker  \  Though  the  hearer 
is  unable  to  give  the  direct  contradictory  of  the  proposi- 
tion, yet  the  speaker  may  propound  the  two  alternatives 

^  According  as  logicians  exclusively  interpret  ^propositions  from 
the  point  of  view  of  the  speaker  (writer)  or  header,  they  are  to  be 
classed  respectively  as  conceptualists  or  nominalists.  The  difference 
between  these  two  points  of  view  lies  at  the  root  of  many  logical 
controversies. 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    169 

that  a  certain  5  is  or  is  not  P,  provided  that  he 
has  his  own  individual  means  of  identifying  the  S  to 
whom  he  is  referring  in  thought.  In  fact  the  most 
common  usage  of  the  phrase  'a  certain'  involves  de- 
liberate concealment  for  various  harmless  purposes  on 
the  part  of  the  speaker.  Thus,  when  I  say  *A  certain 
boy  now  in  this  room  has  stolen  my  purse,'  I  deliber- 
ately preclude  any  hearer  from  strictly  contradicting  or 
agreeing  with  me,  though  of  course  he  could  deny  the 
proposition  by  asserting  not  the  contradictory  but  a 
contrary,  namely:  *No  boy  in  this  room  has  stolen  the 
purse.' 

As  regards  a  narrative,  fictitious  or  historical,  how- 
ever, where  any  substantival  reference  must  always  be 
interpreted  in  accordance  with  its  nexus  with  the  intro- 
ductory *a  certain'  (coupled  or  not  with  a  proper  name), 
the  writer  and  reader  are  so  far  in  the  same  position 
that  neither  the  one  nor  the  other  is  concerned  with 
the  question  of  ultimate  identification.  The  referential 
'the'  is  prefixed  to  an  object  identical  with  that  to  which 
the  introductory  *a'  was  first  prefixed,  but  outside  and 
beyond  this  nexus  there  is  no  further  possibility  of  identi- 
fication. Hence  the  whole  body  of  propositions  in  a  fic- 
titious narrative  is  not  entertained  with  a  view  to  the 
consideration  of  their  truth  or  falsity,  and  might  be  called 
pure  suppositions.  The  scholastic  logicians  introduced 
the  phrase  'suppositio  materialis'  which  would  illustrate 
the  sense  in  which  'supposition'  has  just  been  used; 
but  modern  logicians  have  interpreted  this  phrase  as 
equivalent  to  what  they  call  the  universe  of  discourse.' 
It  is  obvious,  however,  that  the  two  conceptions  are 
totally  distinct,   inasmuch   as   the   former  consists  of 


170 


CHAPTER  X 


classes  of  propositions  included  in  the  universe  of  all  pos- 
sible pro£OsitlQnSj  whereas  the  latter  consists  of  classes 
of  substantives  included  in  the  universe  of  all  possible 
substantives.  In  contrasting  a  work  of  fiction  with  an 
historical  work,  the  propositions  laid  down  in  the  latter 
are  put  forward  as  to  be  accepted  as  true  on  the  authority 
of  the  writer.  But  in  both  cases,  whether  history  or 
fiction,  it  still  holds  that  there  is  no  means  for  ulti- 
mately identifying  the  characters  introduced  either  on 
the  part  of  the  reader  or  the  writer;  we  can  only  say 
that  in  history  it  is  believed  that  these  characters  are 
identifiable  with  persons  who  have  actually  existed, 
whereas  in  fiction  no  such  belief  is  involved. 

§  7.  Within  propositions  which  are  fictitious,  the 
distinctions  between  those  which  introduce  beings  to 
which  there  are,  and  to  which  there  are  not,  similar 
j  beings  in  the  world  of  reality,  gives  rise  to  a  further 
problem\  Thus  we  may  contrast  the  various  statements 
about  the  architect  Mr  Pecksniff  in  Martin  Chuzzlewit 
with  the  various  statements  that  might  be  made  about 
the  fairy  Puck  in  a  fairy-tale.  It  will  be  noted  that  in 
the  former  case  the  general  class  (architect)  to  which 
reference  is  made,  actually  exists,  whereas  in  the  latter 
the  class  (fairy)  to  which  reference  is  made,  does  not 
exist;  while  neither  the  individual  Pecksniff  nor  the 
individual  Puck  does  or  ever  did  exist.  This  imme- 
diately gives  rise  to  the  question  of  the  distinction  or 

*  Of  course  there  is  a  further  distinction  between  fictions  (novels, 
dramas)  which  describe  characters  and  incidents  ^uch  as  might  occur 
in  the  real  world,  and  fairy-tales  (myths,  legends)  which  give  descrip- 
tions such  as  never  can  occur,  in  that  judgments  as  to  their  natural- 
ness or  *  realism'  in  the  former  case  are  more  often  relevant  than  in 
the  latter. 


EXISTENTIAL,  SUBSISTENTIAL  AND  NARRATIVE    171 

relation  in  the  significance  of  the  word  *  exist*  as  applied, 
firstly  to  a  class,  and  secondly  to  an  individual.   We  have 
above  pointed  out  the  peculiar  and  comparatively  modern 
application  of  the  word  *  exist'  on  the  part  of  formal  logi- 
cians, who  express  the  proposition:  'There  are  archi- 
tects' in  the  form  *The  class  architect  exists';  and  the 
proposition  'There  are  no  fairies'  in  the  form  'The  class 
fairy  does  not  exist.'    A  more  precise  formulation  of 
these  propositions  is  obtained  by  taking  C  to  stand 
illustratively  for  any  class,  the  affirmation  of  whose 
existence  is  thus  rendered:  'There  is  at  least  one  indi- 
vidual, say  P,   which  is  comprised  in  C;  or  rather, 
since  a  class  is  determined  by  connotation:  'There  is 
at  least  one  individual,  say  P,  which  is  characterised  by 
the  conjunction  of  adjectives  constituting  the  connota- 
tion of  the  name  C!    Now  here,  I  maintain,  that  the  \ 
symbol  P  stands  for  a  proper  or  uniquely  descriptive 
name,  and  hence  that  the  conception  of  the  existence  [ 
of  a  class — indicated  by  a  connotative  name — requires  j 
the  conception  of  the  existence  of  an  individual — indi- 
cated by  a  proper  or  uniquely  descriptive  name.     Now 
we  may  agree  that  there  is  no  such  individual  as  Peck- 
sniff, and  that  there  is  no  such   individual  as   Puck; 
although  in  the  first  case  the  class  'architect' — which 
might  be  used  in  the  description  of  Pecksniff— would 
be  said  to  exist,  while  the  class  'fairy' — which  might 
be  used   in  the  description  of  Puck — would  be  said 
not  to  exist.     If  then  we  brought  forward  Sir  Christo- 
pher Wren  and  Mr  Pecksniff  as  instances  of  architects, 
or  Oliver  Cromwell  and  Mr  Pecksniff  as  instances  of 
hypocrites,  would  this  substantiate  the  affirmation  that 
there  are  at  least  two  individuals  comprised  in  the  class 


172 


CHAPTER  X 


173 


architect,  or  in  the  class  hypocrite?  If  this  question  is 
answered  in  the  negative  it  must  be  on  the  ground 
that,  in  some  sense  of  the  term  *  exist'  which  is  not  ap- 
propriate to  classes y  Mr  Pecksniff  does  not  and  never 
did  exist,  and  hence  he  cannot  count  as  one  when  we 
are  enumerating  the  members  comprised  in  any  given 
class.  Furthermore,  since  the  numerical  predication  *at 
least  one'  is  highly  indeterminate  and  could  be  in  this 
or  that  case  replaced  by  the  relatively  determinate  'at 
least  n    where  n  stands  for  this  or  that  number,  the 

I  affirmation  that  *the  class  C exists'  is  only  a  special  and 
less  determinate  case  of  the  affirmation  that  'the  class  C 

i  comprises  at  least  n  items,'  and  the  number  n  cannot 
be  counted  as  such  unless  all  the  n  items  exist.  The 
conclusion  therefore  follows  that  the  sense  of  the  word 
'exist'  when  predicated  of  a  class  is  dependent  upon 
that  of  the  word  'exist'  when  predicated  of  an  item  or 
individual  indicated  by  a  proper  or  uniquely  descriptive 
name\ 

I  ^  This  contention  is  directed  against  the  position  held  in  the 
.  Principia  Mathematical  where  E !  is  ultimately  defined  in  terms  of  3, 
i  whereas  in  my  view  3  is  to  be  ultimately  defined  in  terms  of  E ! 


CHAPTER  XI 


THE  DETERMINABLE 


Si.    In  this  chapter  we  propose  to  discuss  a  certain 
characteristic  of  the  adjective  as  such,  which  perhaps 
throws  the  strongest  light  upon  the  antithesis  between 
it  and  the  substantive.    Here  it  will  be  apposite  to  con- 
sider the  traditional  account  of  the  principles  of  logical 
division  where  a  class  (of  substantives)  is  represented 
as  consisting  of  sub-classes.    This  process  is  governed 
by  the  following  rules:    (i)  the  sub-classes  must  be 
mutually  exclusive ;  (2)  they  must  be  collectively  ex- 
haustive of  the  class  to  be  divided;  (3)  division  of  the 
class  into  its  co-ordinate  sub-classes  must  be  based  upon 
someone  'fundamentum  divisionis.'    The  first  two  of 
these  rules  may  be  said  to  be  purely  formal,  and  do 
not  raise  any  problem  of  immediate  interest ;  but  the 
technical  term  fundamentum  divisionis— though    per- 
haps readily  understood  by  the  learner— is  actually  in- 
troduced without  explicit  account  of  its  connection  with, 
or  its  bearing  upon,  ideas  which  have  entered  into  the 
previous  logical   exposition.     To  illustrate  the  notion 
we  are  told,  for  instance,  that,  when  a  class  of  things  is 
to  be  divided  according  to  colour,  or  to  size,  or  to  some 
other  aspect  in  which  they  can  be  compared,  then  the 
colour,  size,  or  other  aspect  constitutes  the  fundamen- 
tum divisionis.    Now  although,  grammatically  speaking, 
words  like  colour  and  size  are  substantival,  they  are  in 


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CHAPTER  XI 


THE  DETERMINABLE 


175 


fact  abstract  names  which  stand  for  adjectives ;  so  that 
the  fundamentum  divisionis  is,  in  the  first  place,  an 
adjective,  and  in  the  second,  an  adjective  of  the  par- 
ticular kind  illustrated  by  'colour'  when  considered  in 
its  relation  to  red,  blue,  green,  etc.  Superficially  this 
relation  appears  to  be  the  same  as  that  of  a  single  object 
to  some  class  of  which  it  is  a  member:  thus  two  such 
propositions  as  'Red  is  a  colour'  and  'Plato  is  a  man' 
appear  to  be  identical  in  form;  in  both,  the  subject  ap- 
pears as  definite  and  singular,  and,  in  both,  the  notion 
of  a  class  to  which  these  singular  subjects  are  referred 
appears  to  be  involved.  Our  immediate  purpose  is  to 
admit  the  analogy,  but  to  emphasise  the  differences 
between  these  two  kinds  of  propositions,  in  which  com- 
mon logic  would  have  said  we  refer  a  certain  object  to 
a  class. 

I  propose  to  call  such  terms  as  colour  and  shape 
determinables  in  relation  to  such  terms  as  red  and  cir- 
cular which  will  be  called  determinates  \  and,  in  intro- 
ducing this  new  terminology,  to  examine  the  distinction 
between  the  relation  of  red  to  colour  and  the  relation 
of  Plato  to  man.  To  predicate  colour  or  shape  of  an 
object  obviously  characterises  it  less  determinately  than 
to  predicate  of  it  red  or  circular-,  hence  the  former 
adjectives  may  be  said  negatively  to  be  indeterminate 

1  compared  with  the  latter.  But,  to  supplement  this  nega- 
tive  account  of  the  determinable,  we  may  point  out  that 

1^.  any  one  determinable  such  as  colour  is  distinctly  other 
than  such  a  determinable  as  shape  or  tone;  i.e.  colour 
is  not  adequately  described  as  indeterminate,  since  it 
is,  metaphorically  speaking,  that  from  which  the  specific 
determinates,  red,  yellow,  green,  etc.,  emanate ;  while 


from  shape  emanate  another  completely  different  series 
of  determinates  such  as  triangular,  square,  octagonal,  etc. 
Thus  our  idea  of  this  or  that  determinable  has  a  distinctly 
positive  content  which  would  be  quite  inadequately  re- 
presented by  the  word  'indeterminate.*    Further,  what 
have  been  assumed  to  be  determinables — e.g.  colour, 
pitch,  etc. — are  ultimately  different,  in  the   important 
i  sense  that  they  cannot  be  subsumed  under  some  one 
higher  determinable,  with  the  result  that  they  are  in- 
comparable with  one  another;  while  it  is  the  essential 
nature  of  determinates  under  any  one  determinable  to 
be  comparable  with  one  another.    The  familiar  phrase 
*  incomparable'  is  thus  synonymous  with  'belonging  to 
different  determinables,'  and  'comparable'  with  'belong- 
ing to  the  same  determinable' ;  not  that  this  is  the  actual 
meaning  of  the  terms,  but  that  enquiry  into  the  reason 
for  the  comparability  or  incomparability  of  two  qualities 
will  elicit  the  fact  that  they  belong  to  the  same  or  to  dif- 
ferent determinables  respectively.  This,  phrase  'belong- 
ing to'  is  also  more  usually  used  of  a  member  of  a  class 
in  relation  to  its  class:  we  have,  then,  to  contrast  the 
significance  of  the  relation  'belonging  to'  when  applied 
in  one  case  to  a  determinate  and  its  determinable,  and 
in  the  other  to  an  individual  and  its  class.    If  it  is  asked 
why  a  number  of  different  individuals  are  said  to  belong 
to  the  same  class,  the  answer  is  that  all  these  different 
individuals  are  characterised  by  some  the  same  adjec- 
tive or  combination  of  adjectives.     But  can  the  same 
reason  be  given  for  grouping  red,  yellow  and  green 
(say)  in  one  class  under  the  name  colour?  What  is  most 
prominently  notable  about  red,  green  and  yellow  is  that 
they  are  different,  and  even,  as  we  may  say,  opponent 


176 


CHAPTER  XI 


THE  DETERMINABLE 


177 


to  one  another ;  is  there  any  (secondary)  adjective  which 
analysis  would  reveal  as  characterising  all  these  differ- 
ent (primary)  adjectives?  In  my  view  there  is  no  such 
(secondary)  adjective;  in  fact,  the  several  colours  are 
put  into  the  same  group  and  given  the  same  name 
colour,  not  on  the  ground  of  any  partial  agreement,  but 
on  the  ground  of  the  special  kind  of  difference  which 
distinguishes  one  colour  from  another ;  whereas  no  such 
difference  exists  between  a  colour  and  a  shape.  Thus  red 
and  circular  are  adjectives  between  which  there  is  no 
relation  except  that  of  non-identity  or  otherness;  whereas 
red  and  blue,  besides  being  related  as  non-identical, 
have  a  relation  which  can  be  properly  called  a  relation 
of  difference,  where  difference  means  more  than  mere 
otherness.  What  is  here  true  of  colour  is  true  of  shape, 
pitch,  feeling- tone,  pressure,  and  so  on:  the  ground  for 
grouping  determinates  under  one  and  the  same  deter- 
minable is  not  any  partial  agreement  between  them 
that  could  be  revealed  by  analysis,  but  the  unique  and 
peculiar  kind  of  difference  that  subsists  between  the 
several  determinates  under  the  same  determinable,  and 
which  does  not  subsist  between  any  one  of  them  and 
an  adjective  under  some  other  determinable.  If  this  is 
granted,  the  relations  asserted  in  the  two  propositions 
*Red  is  a  colour'  and  'Plato  is  a  man,'  xkiow^  formally 
equivalent,  must  yet  be  contrasted  on  the  ground  that 
the  latter  but  not  the  former  is  based  upon  an  adjectival 
predication.  For  the  latter  is  equivalent  to  predicating 
the  adjective  'human'  of  *  Plato/  while,  without  denying 
that  some  adjectives  may  properly  be  predicated  of 
(the  adjective)  red,  yet  the  proposition  *Red  is  a  colour' 
is  not  equivalent  to  predicating  any  adjective  of  red. 


§  2.  Bearing  in  mind  this  distinction,  the  question 
arises  whether  what  are  called  abstract  names  can  be 
divided  in  the  same  way  as  concrete  names  into  smgular 
and  general ;  in  other  words,  whether  adjectives  can  be 
divided  into  these  two  classes.  The  answer  seems  to  be 
that  adjectives  can  be  divided  into  two  classes  more  or 
less  analogous  to  the  singular  and  general  which  dis- 
tinguish substantives,  but  that  the  two  different  kinds  of 
adjectives  are  preferably  distinguished  as  determinate 
and  indeterminate.  When,  in  considering  difterent  de- 
grees of  determinateness,  the  predication  of  one  adjective 
is  found  to  imply  another,  but  not  conversely,  then  the 
former  we  shall  call  a  super-determinate  of  the  latter 
and  the  latter  a  sub-determinate  of  the  former.  Thus 
the  relation  of  super-determinate  to  sub-determinate 
means  not  only  that  the  former  is  more  determinate  than 
the  latter,  but  also  that  the  predication  of  the  former 
would  imply  that  of  the  latter.  A  simple  example  can 
be  taken  from  the  determinable  'number':  thus  7  is 
super-determinate  to  'greater  than  3';  the  adjective 
'greater  than  3,'  though  not  itself  a  summunt  determin- 
able, may  be  called  determinable,  inasmuch  as  it  is  not 
merely  indeterminate  but  capable  of  being  further  de- 
termined in  the  sense  that  it  generates  a  definite  series 
of  determinates.  To  illustrate  more  precisely  what  is 
meant  by  'generates' ;  let  us  take  the  determinable  'less 
than  4 ' ;  then  '  less  than  4 '  generates  '  3 '  and  '  2 '  and  '  i ' 
in  the  sense  that  the  understanding  of  the  meaning  of 
the  former  carries  with  it  the  notion  of  the  latter.  Now  | 
no  substantive  class-name  generates  its  members  in  this  j 
way;  take,  for  instance,  'the  apostles  of  Jesus,'  the  under- 
standing of  this  class-name  carries  with  it  the  notion 

J.  L. 


12 


178 


CHAPTER  XI 


THE  DETERMINABLE 


179 


'men  summoned  by  Jesus  to  follow  him,'  but  it  does 
not  generate  'Peter  and  John  and  James  and  Matthew 
etc.,'  and  this  fact  constitutes  one  important  difference 

i  between  the  relation  of  sub-determinate  to  super-deter- 
minate adjectives  and  that  of  general  to  singular  sub- 
stantives. 

§  3.  Another  equally  significant  difference  is  brought 
out  by  considering  that  aspect  of  substantive-classes  in 
which — to  use  the  terminology  of  formal  logic — increase 

i  of  intension  is  accompanied  by  decrease  of  extension. 
The  phrase  'increase  of  intension'  conjures  up  the 
notion  of  adding  on  one  attribute  after  another,  by  the 
logical  process  called  conjunction ;  so  that,  taking/,  ^,  r, 
to  be  three  adjectives,  increase  in  intension  would  be 
illustrated  by  regarding  /,  ^,  r  conjoined  as  giving  a 
greater  intension  than  /,  ^ ;  and  /,  ^  as  giving  greater 
intension  than  p.  We  have  now  to  point  out  that  the 
increased  determination  of  adjectival  predication  which 
leads  to  a  narrowing  of  extension  may  consist — not  in 
a  process  of  conjunction  of  separate  adjectives — but  in 
the  process  of  passing  from  a  comparatively  indeter- 
minate adjective  to  a  comparatively  more  determinate 
adjective  under  the  same  determinable.  Thus  there  is 
a  genuine  difference  between  that  process  of  increased 
determination  which  conjunctivally  introduces  foreign 
'  adjectives,  and  that  other  process  by  which  without  in- 
creasing, so  to  speak,  the  number  of  adjectives,  we  define 
them  more  determinately. 

In  fact,  the  foreign  adjective  which  appears  to  be 
added  on  in  the  conjunctive  process,  is  really  not  intro- 
duced from  outside,  but  is  itself  a  determinate  under 
another  determinable,  present  from  the  start,  though 


suppressed  in  the  explicit  connotation  of  the  genus. 
We  propose  to  use  a  capital  letter  to  stand  for  a  de- 
terminable, and  the  corresponding  small  letter  with 
various  dashes  to  stand  for  its  determinates.  Thus, 
in  passing  from  the  genus  p  to  the  species  /^,  we  are 
really  passing  (rompQ  to  pq\  or  again  the  apparent  in- 
crease of  intension  from  p  to  pq  to  pqr  is  more  correctly 
symbolised  as  a  passing  ixova  pQR  to  pqR  X.o pqr.  In 
the  successive  process  of  dividing  a  summum  genus  into 
the  next  subordinate  sub-genera,  and  this  again  into 
sub-sub-genera,  the  summum  genus  ought  to  be  repre- 
sented by  a  conjunction  of  determinables,  say  PQRST\ 
the  genera  next  subordinate  to  this,  hypQRST.p'QRST, 
p"QRST,  etc.,  and  the  genera  next  subordinate  to  the 
first  of  these  hy  pqRST,  Pq'RST,  pq'^RST,  and  so  on 
down  to  the  infima  species  represented  by  determinates. 
Thus: 

PQRST 


pQRST       p'QRST       p'QRST. 


PqRST       Pq'RST       Pq"  RST 

In  this  way  we  represent  from  the  outset  the  nature 
of  the  ultimate  individuals  under  the  summum  genus, 
as  being  characterisable  jointly  by  the  determinables 
PQRST,  while  any  genus  or  species  is  represented  by 
these  same  determinables,  one  or  more  of  which  are 
replaced  by  determinates.  This  meets  a  criticism  which 
has  often  been  directed  against  the  formal  account  of 
the  inverse  variation  of  extension  and  intension,  since 
we  see  now  that  the  same  number  of  adjectives  should 
be  used  in  giving  the  connotation  of  the  wider  as  of  the 
narrower  class.  To  illustrate  these  symbols  from  a 
botanical  classification  of  plants :  let  the  determinable 


12 — 2 


/ 


Sf 


i8o 


CHAPTER  XI 


THE  DETERMINABLE 


i8i 


P  stand  for  the  number  of  cotyledons,  Q  for  the  dis- 
position of  the  stamens,  R  for  the  form  of  the  corolla, 
S  for  the  attachment  of  the  petals  and  sepals,  and  T  for 
the  divisibility  of  the  calyx.  Then  PQRST  represents 
the  summum  genus  'plants  as  describable  under  these 
five  heads,  but  otherwise  undetermined  in  character. 
Then/, /',/^  might  stand  respectively  for  having  no 
cotyledons,  having  i,  and  having  2,  thus  representing 
the  defining  characteristic  of  each  of  the  three  classes — 
acotyledon,  monocotyledon,  and  dicotyledon — by  the 
symbols /eT?^^,  p'QRST,  fQRST.  Again  q,  q\  q'\ 
might  stand  respectively  for  the  stamens  being  under, 
around  or  upon  the  carpels,  thus  representing  the  three 
sub-divisions — hypogynous,  perigynous,  epigynous — of 
dicotyledons,  2Ls/^qRST,  fq'RST,fq"RST,  Taking 
regular  and  irregular  to  be  the  two  possible  forms  of 
corolla,  then  the  next  sub-division  under /'y^5  7"  will 
be  fq'rST  and  fq'r'ST,  Again  s  and  /  may  stand 
respectively  for  separability  and  inseparability  of  the 
calyx  and  corolla,  and  yield  the  further  sub-divisions, 
sdcy  p^'q'rsT,  p^'q'rs'T,  The  calyx  may  contain  only  one 
part  or  3  or  4  or  5  or  6,  and  if  these  are  represented 
respectively  by  t,  t\  t'\  t"\  f'\  a  relatively  determinate 
characterisation  is  finally  symbolised  hy  p'^q'rs't'^'  say. 

There  are  cases  for  which  a  modification  of  this 
general  scheme  is  required.  The  cases  are  those  in 
which  one  particular  sub-division  is  definable  by  the 
absence  of  an  element  upon  which  the  predication  of 
other  determinables  depend,  while  in  the  sub-divisions 
co-ordinate  with  this  the  element  in  question  is  present. 
For  example,  the  class  of  plants  called  acotyledons 
might  be  defined  by  the  absence  of  any  corolla,  etc.,  and 


hence  such  variations  as  that  of  the  form  of  the  corolla 
or  the  disposition  of  the  stamens,  etc.,  are  inapplicable 
to  this  particular  sub-division. 

S  4.  Now  adjectives  under  the  same  determinable 
are  related  to  one  another  in  various  ways;  One  rela- 
tional characteristic  holds  in  all  cases ;  namely  that,  if 
any  determinate  adjective  characterises  a  given  sub- 
stantive, then  it  is  impossible  that  any  other  determi- 
nate under  the  same  determinable  should  characterise 
the  same  substantive:  e.g.  the  proposition  that  'this 
surface  is  red'  is  incompatible  with  the  proposition  'this 
(same)  surface  is  blue.'  It  has  been  usual  to  modify  the 
above  statement  by  adding  the  qualification — at  the 
same  time  and  at  the  same  place;  this  qualification 
applies  where  the  substantive  extends  through  some 
period  of  time  and  over  some  region  of  space,  in  which 
case  the  existent  substantive,  having  temporal  or  spatial 
parts,  may  be  said  to  be  extended.  For  this  reason 
the  qualification  would  perhaps  better  be  attached  to 
the  substantive  itself,  and  we  should  say  that,  where 
opponent  adjectives  are  predicated,  reference  is  made  to 
different  substantives,  since  any  one  part  of  an  extended 
substantive  is  existentially  other  than  any  other  part. 

A  second  characteristic  of  many  determinates  under 
the  same  determinable  is  that  the  differences  between 
different  pairs  of  determinates  can  be  compared  with 
one  another;  so  that  if  a,  b,  c,  are  three  determinates, 
there  are  cases  in  which  we  may  say  that  the  differ- 
ence between  a  and  c  is  greater  than  that  between  a 
and  b\  e.g.  the  difference  between  red  and  yellow  is 
greater  than  that  between  red  and  orange.  In  this 
case  the  several  determinates  are  to  be  conceived  as 


l82 


CHAPTER  XI 


THE  DETERMINABLE 


183 


necessarily  assuming  a  certain  serial  order,  which  de- 
velops from  the  idea  of  what  may  be  called  *  adjectival 
betweenness/  The  term  'between'  is  used  here  in  a  fa- 
miliar metaphorical  sense  derived  from  spatial  relations, 
and  is  figuratively  imaged  most  naturally  in  spatial  form. 
Thus  if  b  is  qualitatively  between  a  and  c,  and  c  qualita- 
tively between  b  and  ^,and  so  on,  the  whole  series  has  its 
order  directly  determined  by  the  nature  of  the  adjectives 
themselves.  The  further  distinctions  amongst  series  as 
interminable  or  as  cyclic,  and  again  of  series  of  more 
than  one  order  of  dimensions,  lead  to  logical  complexi- 
ties which  need  not  be  entered  into  here.  Suffice  it  to 
say  that  this  characteristic,  which  holds  of  so  many 
j  determinates,  gives  significance  to  another  well-known 
I  rule  for  logical  division:  divisio  non  facial  sallum:  one 
meaning  of  which  appears  to  be  that  we  contemplate 
not  merely  enumerating  a  set  of  coordinate  sub-classes, 
but  enumerating  them  in  a  certain  order.  The  rule  pre- 
scribes that  the  order  in  which  the  sub-classes  are  enu- 
merated should  correspond  to  the  order  of 'betweenness' 
predicable  of  their  differentiating  characteristics. 

The  order  of  betweenness  which  characterises  the 
determinates  just  considered  may  be  either  discrete  or 
continuous.  I  n  the  case  of  discrete  series  there  is  one  de- 
terminate that  can  be  assigned  as  next  after  any  given 
determinate;  but,  in  the  case  of  a  continuous  series,  a 
determinate  can  always  be  conceived  as  between  any 
two  given  determinates,  so  that  there  are  no  two  deter- 
minates which  can  be  said  to  be  next  to  one  another  in 
the  serial  order.  It  follows  from  this  account  of  continuity 
that,  between  any  twQ  determinates  which  may  be  said 
to  have  a  finite  adjectival  difference,  may  be  interpolated 


an  indefinite  number  of  determinates  having  a  finite  dif- 
ference, and  this  number  becomes  infinite  as  the  differ- 
ences become  infinitesimal.  Amongst  continuous  series 
further  differences  between  the  interminable  and  the 
cyclic,  and  again  between  those  of  one  or  more  order 
of  dimensions,  hold  as  in  discrete  series. 

The  reference  here  to  determinates  of  higher  or  1 
lower  dimension  requires  explanation.  Our  familiar  ex- 
ample of  colour  will  explain  the  point:  a  colour  may 
vary  according  to  its  hue,  brightness  and  saturation ;  so 
that  the  precise  determination  of  a  colour  requires  us 
to  define  three  variables  which  are  more  or  less  inde- 
pendent of  one  another  in  their  capacity  of  co-variation ; 
but  in  one  important  sense  they  are  not  independent  of 
one  another,  since  they  could  not  be  manifested  in  se- 
paration. The  determinable  colour  is  therefore  single, 
though  complex,  in  the  sense  that  the  several  consti- 
tuent characters  upon  whose  variations  its  variability 
depends  are  inseparable. 

§  5.  Returning  to  the  conception  of  the  absolutely 
determinate  adjective,  we  have  to  note  an  important 
distinction  between  absolutely  determinate  and  com- 
paratively indeterminate  predications.  The  distinction 
may  thus  be  formulated:  If,  of  two  substantives  the  a. 
same  determinate  adjective  can  be  predicated,  then  all 
the  adjectives  and  relations  definable  in  terms  of  the 
determinable,  that  can  be  predicated  of  the  one,  could 
be  predicated  of  the  other.  But  if,  of  two  substantives  p_. 
the  same  indeterminate  adjective  can  be  predicated,  then 
only  certain  of  the  adjectives  and  relations  definable  in 
terms  of  the  determinable,  that  can  be  predicated  of 
the  one,  can  be  predicated  of  the  other.    To  illustrate 


i84 


CHAPTER  XI 


THE  DETERMINABLE 


185 


first  the  case  of  an  indeterminate  predication;  let  us 
take  the  numerical  adjective  'greater  than  7';  then  of 
any  collection  of  which  this  numerical  adjective  could 
be  predicated,  other  adjectives  such  as  'greater  than  5' 
and  'greater  than  3 'could  also  be  predicated;  but  some 
collections  that  are  'greater  than  7'  such  as  the  apostles, 
are  greater  than  1 1  and  divisible  by  4  for  instance, 
whereas  other  collections  that  are  'greater  than  7,'  such 
as  the  muses,  are  less  than  1 1  and  are  not  divisible  by 
4:  hence  it  is  only  some  of  the  numerical  adjectives  that 
are  predicable  of  the  muses  that  are  also  predicable  of 
the  apostles,  although  the  adjective  'greater  than  7'  is 
predicable  of  them  both.  Turning  now  to  the  case  of 
dete^^minate  predication;  if,  instead  of  defining  a  col- 
lection by  the  indeterminate  adjective  'greater  than  7,' 
we  had  defined  it  by  the  determinate  adjective  'twelve,' 
then  any  numerical  adjective  that  is  predicable  of  one 
collection  of  twelve,  say  the  apostles,  would  be  predic- 
able of  any  other  collection  of  twelve,  say  the  months 
of  the  year  or  the  sons  of  Israel;  for  example,  'greater 
than  II,'  'divisible  by  4,'  'a  factor  of  96/  What  we 
have  here  seen  to  hold  of  determinate  and  indetermi- 
nate number  holds  of  any  other  determinable.  The  case 
of  colour  lends  itself  easily  for  illustration  on  account  of 
the  specific  names  which  have  been  assigned  to  its 
determinates :  thus,  if  the  colours  of  two  different  objects 
are  the  same  shade  of  yellow,  then  though  these  two 
objects  may  differ  in  any  number  of  other  respects  such 
as  shape  and  size,  yet  we  may  say  that  any  colour- 
property  of  the  one  object  will  agree  with  the  colour- 
property  of  the  other;  if  the  colour  of  one  is  more 
brilliant  or  less  saturated  than  the  colour  of  an  orange, 


then  the  same  will  hold  for  the  colour  of  the  other. 
In  fact,  whatever  sensational  determinable    we   take, 
whether  it  be  colour,  or  sound,  or  smell,  the  determinate 
characterisations  under  any  such  determinable  would 
lead  to  the  same  forms  of  generalisation  that  have  been 
developed  by  science  only  in  the  sphere  of  quantity. 
It  is  agreed  that  in  the  sphere  of  sense  perception,  dif- 
ferences of  quality  strictly  speaking  hold  only  of  the 
mental  or  sensational,  and  that  the  physical  can  only 
be  defined  in  quantitative  terms.    Thus  in  the  Weber- 
Fechner  experiments  the  experient  judges  of  equiva- 
lence or  difference  in  the  intensity  or  quality  of  his 
sensations,  with  which  are  correlated  quantitative  dif- 
ferences in  the  stimuli.    The  attempts  that  psycholo- 
o-ists  have  made  to  discover  formulae  of  correlation 
between  the  stimuli  on  the  one  hand  and  the  sensations 
on  the  other  hand  show  that  determinateness  in  a  quali- 
tative or  intensive  scale  carries  with  it  the  same  logical 
consequences  as  does  determinateness  of  magnitude  for 
physically  measurable  quantities.     Furthermore  deter- 
minateness in  either  case  is  only  approximately  attain- 
able, whether  we  rely  upon  the  immediate  judgments  of 
perception  or  are  able  to  utilize  instruments  of  measure- 
ment.   The  practical  impossibility  of  literally  determi- 
nate characterisation  must  be  contrasted  with  the  uni- 
versally adopted  postulate  that  the  characters  of  things 
which  we  can  only  characterise  more  or  less  indetermi- 
nately, are,  in  actual  fact,  absolutely  determinate'. 

'  The  notion  of  the  Determinable  will  be  shown  in  later  chapters 
to  have  importance  in  a  large  number  of  applications. 


i86 


THE  RELATION  OF  IDENTITY 


187 


CHAPTER  XII 

THE  RELATION  OF  IDENTITY 

§  I.  The  occasion  for  using  the  relation  of  identity 
is  where  a  common  term  appears  in  different  connec- 
tions ;  thus  we  use  the  idea  of  identity  always  along 
with  the  idea  of  difference.  The  logical  relation  between 
difference — or  more  properly  otherness — and  identity, 
is  that  of  co-opponency :  that  is,  taking  A  and  B  as 
any  two  terms,  it  cannot  be  that  A  is  both  identical 
with  and  other  than  B,  and  it  must  be  that  A  is  either 
identical  with  or  other  than  B,  Thus  the  relation 
between  identity  and  otherness  is  reciprocal.  It  must 
therefore  be  explained  that  we  cannot  define  otherness 
as  meaning  non-identity  any  more  than  we  can  define 
identity  as  meaning  non-otherness.  The  conceptions 
of  identity  and  of  otherness  must  be  separately  and 
independently  understood  before  we  can  assert  the 
above  axioms. 

The  most  trivial  and  apparently  insignificant  use  of 
the  relation  of  identity  is  expressed  in  the  formula  '  x 
is-identical-with  x',  where  what  is  primarily  meant  is 
that  in  repeated  occurrences  of  the  word  x,  either  in 
a  special  context  or  irrespective  of  context,  the  word 
shall  mean  in  any  later  occurrence  what  it  meant  in  an 
earlier  occurrence.  Thus,  even  in  this  very  elementary 
usage,  the  idea  of  identity  goes  along  with  the  idea  of 
otherness ;   for  identity  applies  to  what  is  meant  by 


the  word,  and  otherness  to  its  several  occurrences. 
Underlying  this  characteristic  of  language  there  is  the 
corresponding  characteristic  of  thought ;  thus,  in  using 
'X  is-identical-with  x'  in  reference  to  entities  and  not 
mere  words,  we  return  in  thought  to  the  object  pre- 
viously thought  of,  so  that  identity  applies  to  what 
constitutes  the  object  of  thinking,  and  otherness  to  the 
several  recurrent  acts  of  thought. 

A  less  elementary  usage  of  the  relation  of  identity 
occurs  in  the  definition  of  words  or  phrases ;  thus,  if  x 
and  y  stand  for  two  different  phrases,  we  may  speak  of 
X  being  identical  with  y,  although  the  phrases  are 
palpably  different.  Here  otherness  applies  to  the 
phrases  and  identity  to  what  is  meant  by  the  phrases. 
Verbal  identification  of  x  with  y  may  be  contrasted 
with  factual  identification  ;  here  the  relation  of  identity 
applies  (as  before)  to  the  objects  denoted  by  the 
words ;  but  the  proposition  asserting  identity  in  the 
one  case  is  of  a  different  nature  from  the  proposition 
asserting  identity  in  the  other  case :  for  in  the  first 
case  it  is  verbal,  in  the  second  factual.  The  relation  of 
identity  asserted  in  the  two  propositions :  '  Courage  is 
the  mean  between  timidity  and  foolhardiness '  and 
*  Courage  is  the  one  virtue  required  of  a  soldier '  is  the 
same  :  but  the  natures  of  the  propositions  differ,  since 
the  first — being  put  forward  as  a  definition — is  verbal, 
and  the  second  is  factual.  More  generally,  we  may 
distinguish  the  different  grounds — such  as  rational,  ex- 
periential or  linguistic — upon  which  the  assertion  of  any 
specific  logical  relation  is  based ;  but  these  differences  in 
the  grounds  of  assertion,  do  not  affect  the  nature  of  the 
relation  asserted.  Thus,  abbreviating  *  is  identical  with  * 


r>i 


i88 


CHAPTER  XII 


THE  RELATION  OF  IDENTITY 


189 


i 


into  '  i,'  from  the  verbal  statement  xiy  together  with  the 
factual  statement  jv^>  we  may  correctly  infer  the  factual 
statement  xiz.  This  inference  uses  the  transitive^  pro- 
perty of  identity,  and  would  therefore  be  impossible 
unless  the  relation  of  identity  asserted  in  a  verbal 
statement  was  the  same  as  that  asserted  in  a  factual 
statement. 

§  2.  We  have  shown  the  proper  sense  in  which 
difference  can  be  said  to  be  involved  in  identity,  but 
many  philosophers  have  laid  down  the  dictum  that 
identity  implies  difference,  in  the  sense,  apparently, 
that  when  we  assert  that  A  is  identical  with  B  we 
are  also  involved  in  the  assertion  that  A  is  different 
from  B,  The  plausibility  of  this  dictum  depends  upon 
a  certain  looseness  in  the  application  of  the  word  *  un- 
plies ' ;  thus  the  statement  that  identity  implies  difference 
is  correct  in  the  sense  that  the  asserting  of  identity 
between  one  pair  of  terms  implies  our  having  implicidy 
or  tacitly  asserted  difference  between  another  pair  of 
terms.  This  follows  from  what  has  been  said  above ; 
e.g.  when  identifying  the  colour  of  this  with  the 
colour  of  that,  we  are  implicitly  differentiating  'this' 
from  'that';  and  thus  the  identification  and  the  dit- 
ferentiation  may  properly  be  said  to  be  component 
parts  of  a  single  mental  act.  But,  to  give  a  more  pre- 
cise statement  of  this  implication,  it  would  be  necessary 
to  say  that,  when  A  is  identical  with  B  in  a  certahi 
respect,  then  A  is  different  from  B  in  some  other  respect. 
In  common  language  it  is  of  course  perfectly  legitimate 
to  say  of  two  things  that  they  are  identical  in  respect 

^  For  the  term  *  transitive'  as  a  property  of  the  relation  of  identity 
see  Part  1,  Chapter  XIV. 


of  colour  and  different  in  respect  of  shape.  But  here 
the  term  '  identity '  should  be  used  more  precisely  :  we 
ought  properly  to  say  that,  while  the  colour  of  this  is 
identical  with  the  colour  of  that,  the  shape  of  this  is 
different  from  the  shape  of  that.  When  then  identity 
and  difference  go  together  in  this  way  we  ought  to  say, 
not  that  the  things  are  both  identical  and  different,  but 
that  one  of  their  qyiialities  is  identical  and  another 
different.  To  complete  the  account  of  the  modes  in 
which  identity  and  difference  mutually  involve  one 
another  without  confusion,  we  need  only  take  four  such 
typical  elementary  propositions  as  : 

(i)    this  is/, 

(ii)    this  is  q, 

(iii)   that  is/, 

(iv)    that  is  q, 

where  /  and  q  stand  for  different  qualities,  and  '  this ' 
and  'that'  for  different  things.  Then,  comparing  (i) 
with  (ii)  or  (iii)  with  (iv),  we  have  identity  of  thing  and 
otherness  of  quality ;  and,  comparing  (i)  with  (iii)  or 
(ii)  with  (iv),  we  have  otherness  of  thing  and  identity 
of  quality  ;  finally,  comparing  (i)  with  (iv)  or  (ii)  with 
(iii),  we  have  both  otherness  of  thing  and  otherness  of 

quality. 

S  \,  The  relation  and  distinction  between  thing 
and  quality  may  be  generalised  in  terms  of  the  correla- 
tive notions  of  substantive  and  adjective,  the  latter 
admitting  of  further  resolution  into  determinates  and 
their  determinables.  Thus,  when  predicating  the  same 
adjective/  of  this  and  of  that  substantive,  we  shall  say 
that  this  and  that  agree  as  regards  the  determinable 


IQO 


CHAPTER  XII 


THE  RELATION  OF  IDENTITY 


191 


P\  and  when  predicating  /  of  this  and  /'  of  that 
substantive,  we  shall  say  that  this  and  that  disagree  as 
regards  the  determinable  P.  Here  the  term  '  disagree ' 
is  used  in  place  of  'differ,'  for,  strictly,  speaking,  'dif- 
ference '  is  a  relation  between  adjectives  under  the 
same  determinable  ;  and  in  measuring  different  degrees 
of  difference  amongst  such  adjectives,  we  may  speak 
of  the  substantives  as  being  similar  when  the  degree 
of  difference  between  the  adjectives  characterising 
them  is  small,  and  as  being  ddssimilar  when  the  degree 
of  difference  is  great.  Further  we  may  say  that  two 
substantives  partially  disagree  when  they  are  charac- 
terised by  the  san^  determinates  under  certain  deter- 
minables  and  by  different  determinates  under  certain 
I  other  determinables.  But  partial  agreement  must  be 
;  distinguished  from  approximate  agreement,  otherwise 
\  called  simnarity ;  and  partial  disagreement  must  be 
distinguished  from  remote  disagreement,  otherwise 
called  dissimilarity.  The  distinction  between  similarity 
and  dissimilarity  involves  reference  to  adjectives  under 

et,   the  same  determinable,  and  is  obviously  a  matter  of 
degree ;  while  partial  agreement  or  disagreement  in- 

'^.   volves  reference  to  adjectives  under  different  deter- 
minables. 

§  4.  Adjectives  under  the  same  determinable  are 
usually  said  to  be  comparable  ;  whereas  those  under 
different  determinables  are  said  to  be  incomparable  or 
disparate.  This  point  raises  a  question  of  considerable 
psychological  interest  as  to  the  possibility  of  comparing 
two  such  disparate  characters  (say)  as  red  and  a  trumpet- 
blast.  The  possibility  of  such  comparison  may  perhaps 
be  accounted  for  by  association ;  or  it  may  be  that  some 


real  deep-rooted  form  [of  connection  underlies  the  two 
characters,  which,  if  it  could  be  explicitly  rendered, 
would  have  its  logical  place  amongst  relations  such  as 
those  which  we  are  now  discussing.    But  apart  from 
this  possible  topic  of  psychological  interest,  it  is  not 
usual    to   speak    of   '  red '    and    *  a    trumpet-blast '    as 
being  either  like  or  unlike  ;   and  it  is  more  usual  to 
restrict  the  use  of  the  terms  like  and  unlike  to  qualities 
belonging  to  the  same  determinable,  such  as  colour  or 
sound.     If  this    is   admitted,  the    conclusion  at  once 
follows  that  like  and  unlike  are  not  proper  logical  con- 
tradictories, but  are  relations  of  degree  \  so  that  to  say 
of  two  comparable  qualities  that  they  are  more  or  less 
like  is  equivalent  to  saying  that  they  are  less  or  more 
unlike ;  we  cannot  define  the  point  at  which  the  rela- 
tion of  difference  changes  from  likeness  to  unlikeness, 
their  opposition  being  only  one  of  degree.   When  we 
compare  different  degrees  of  difference  between  deter- 
minates under  a   determinable   whose  variations  are 
continuous,  and  judge,  for  instance,  that  the  difference 
between  A  and  C  is  greater  than  that  between  A  and 
B,  such  differences  between  the  determinates  may  be 
said  to  have  distensive  magnitude.   When  this  distensive 
magnitude   is  too  small,  we  fail    perceptually  to  dis- 
criminate (say)  between  A  and  B ;  and  some  psycho- 
logists have  virtually  taken  *  identity '  to  be  equivalent, 
in  such  cases,  to  minimum  discernible  difference.    This, 
however,  entails  logical  contradiction  ;  for  the  concep- 
tion of  a  *  minimum  discernible  difference  '  implies  that 
we  fail  to  discriminate  between  qualities,  which  really 
are  different  and  not  identical ;  and  that  strict  identity 
can  only  be  predicated  when  difference  has  reached  the 


192 


CHAPTER  XII 


THE  RELATION  OF  IDENTITY 


193 


absolute  limit  zero.  On  this  ground,  it  might  be  urged 
that  identity  when  applied  to  qualities  susceptible  of 
continuous  variation  means  zero  difference  ;  and  has, 
therefore,  a  different  significance  from  identity,  when 
applied  to  things  or  to  qualities  which  vary  discretely. 
This  contention  would,  in  my  view,  be  fallacious  ;  for 
it  w^ould  appear  to  involve  a  confusion  between  the 
objective  conception  of  identity  itself  and  the  subjec- 
tive limitations  in  our  power  of  judging  identity.  On 
the  other  hand,  difference  when  applied  to  adjectives 
under  the  same  determinable  has  a  certain  meaning 
which  is  distinct  from  any  meaning  of  difference  ap- 
plicable to  substantives  or  to  adjectives  under  one  and 
another  determinable.  As  regards  the  latter,  difference 
can  only  mean  mere  otherness ;  but  as  regards  the 
former,  difference  may  mean  more  than  mere  otherness  ; 
viz.  something  that  can  be  measured  as  greater  or 
smaller.  Thus  Socrates  is  merely  other  than  Plato,  red 
is  merely  other  than  hard  ;  but  round  and  square,  red 
and  yellow,  five  and  nine  are  not  merely  non-identical, 
but  are  also  such  that  the  difference  between  them  can 
be  apprehended  as  greater  or  smaller  (say)  than  that 
between  oblong  and  square,  orange  and  yellow,  seven 

and  nine. 

§  5.  There  is  yet  another  aspect  of  the  dictum: 
identity  (of  adjective)  implies  difference  (of  substan- 
tive) according  to  which  it  could  equally  well  be  ren- 
dered: difference  (of  adjective)  implies  difference  (of 
substantive).  For,  where  identity  applies  to  the  adjec- 
tive and  difference  to  the  substantive,  identity  may 
properly  be  said  to  imply  difference,  in  the  sense  that 
the  identity  predicated  of  an  adjective  is  used  along 


with  otherness  as  predicated  of  compared  substantives ; 
but,  in  this  sense,  we  may  also  say  that  difference  im- 
plies difference;  i.e.  that  difference  predicated  between 
two  adjectives  is  used  along  with  otherness  as  predi- 
cated of  the  compared  substantives.  The  dictum  should 
therefore  be  expressed  in  more  general  terms  to  include 
identity  and  difference  in  respect  of  the  adjectives  char- 
acterising one  and  another  substantive.  Thus:  Com- 
parison with  respect  to  any  determinable  character, 
whether  it  yields  identity  or  difference,  presupposes 
otherness  of  the  substantives  characterised  by  the  de- 
terminable in  question.  In  this  connection  we  may 
examine  the  contrast  commonly  drawn  between  quali- 
tative difference  and  numerical  difference.  This  termin- 
ology is  incorrect,  for  'numerical  difference'  simply 
means  otherness —the  very  notion  *  numerical'  owing  its 
origin  to  the  conception  of  mere  otherness,  which  is  the 
basis  of  number.  Again  in  contrasting  qualitative  with 
numerical  difference  there  is  the  suggestion  that  other- 
ness does  not  apply  to  qualities  or  adjectives,  whereas 
in  its  developments  into  number  otherness  is  clearly  seen 
to  apply  precisely  in  the  same  way  to  adjectives  as  to 
substantives.  In  our  view  the  required  distinction  is  that 
which  was  drawn  above  between  the  word  difference  as  *. 
meaning  merely  otherness,  and  the  word  difference  in  ^ 
its  exclusive  application  to  adjectives  under  the  same 

determinable. 

§  6.     Under  the  head  of  difference  and  otherness  a 
special  problem  to  be  discussed  is  that  involved  in  the   ^ 
famous  Leibnizian  principle  'tjie  identity  of  indiscern- 
ibles.'    This  phrase  signifies  that  'indiscernibility  im- 
plies identity,'  which  is  an  awkward  way  of  saying  that 

j.L.  13 


194 


CHAPTER  XII 


THE  RELATION  OF  IDENTITY 


195 


>/ 


y 


*  otherness  implies  discernibility.'    Here  the  term  dis- 
cernibility  has  not  a  psychological  but  a  purely  onto- 
logical  significance.     More  explicitly,  the  phrase  sig- 
nifies that  a  plurality  of  existent  objects  is  only  possible 
so  far  as  there  is  some  difference  in  the  qualities  or 
relations  which  can  be  predicated  of  them.     If,  in  this 
phrase,  the  term  relation  is  interpreted  to  include  such 
external  relations  as  space  or  time,  then  no  reasonable 
criticism  of  the  Leibnizian  formula  could  be  maintained; 
for,  as  has  been  contended  from  another  point  of  view 
\  in  Chapter  II,  'existential  otherness'  implies  difference 
^  in  spatial  or  temporal  relations.    But  this  interpretation 
can  hardly  be  taken  to  represent  Leibniz's  meaning,  since 
he  denied  external  relations,  and  held  that  this  denial 
demonstrates  the  non-reality  of  space.    But  whereas  he 
pretends  to  base  the  denial  of  space  upon  his  dictum,  in 
t  reality  his  dictum  would  have  no  plausibility  unless  it 
I  had  been  previously  agreed  that  space  was  unreal.  What 
Leibniz  certainly  meant  was  that  two  existent  objects 
could  not  agree  in  all  their  internal  characters  and  rela- 
tions.  The  difficulty  that  here  arises  in  regard  to  the 
number  of  respects  and  the  remoteness  of  difference 
that  are  abstractly  necessary  for  the  possibility  of  two- 
ness  of  existence  exhibits  more  emphatically  the  purely 
dogmatic  nature  of  the  Leibnizian  principle,  which  seems 
to  me  in  any  case  to  have  no  logical  justification  what- 
ever\ 


'  Much  the  same  considerations  were  brought  forward  in  the 
criticism  of  Bradley's  dictum  that  *  distinction  implies  difference' 
(see  Chapter  II).  If  the  Leibnizian  and  the  Bradleyan  principles 
can  be  in  any  way  distinguished,  it  is  that  the  former  is  ontological 
and  the  latter  epistemological. 


§  7.     Having  discussed  the  notion  of  identity  in  its 
contrasts  and  connections  with  difference  and  otherness, 
we  must  finally  examine  the  nature  of  the  relation  of 
identity  itself,  apart  from  its  connections  with  other  re- 
lations.   The  problem  may  be  indicated  by  discussing  j 
whether  identity  is  or  is  not  definable.    For  this  purpose ' 
it  will  be  desirable  to  begin  by  considering  the  formula 
xix,  rather  than  xiy.    For  xiy  can  only  be  interpreted 
by  explicitly  distinguishing  the  word,  phrase  or  symbol 
which  denotes  an  entity  from  the  entity  itself  that  is 
denoted ;  and  moreover  the  proposition  xiy  (until  defi- 
nite equivalents  for  x  and  y  are  substituted)  can  only  be 
adopted  hypothetically  or  illustratively.    On  the  other 
hand,  the  straightforward  proposition  xix  is  to  be  as- 
serted on  rational  grounds  for  any  value  of  x.     Elimi- 
nating the  symbol  x,  what  is  to  be  universally  asserted  is 
that  'Any  entity  is  identical  with  itself.'  Unfortunately 
the  term  itself  z2Si  only  be  defined  as  '  what  is  identical 
with    i£\    and  hence  any  explication  of  the  formula 
seems  to  lead  to  an  infinite  regress.    This  difficulty  can 
be  removed  by  expressing  the  formula  in  the  negative 
form:    *No  entity  is   identical  with  any  entity  other 
than  itself.'    This  is  to  be  understood  as  a  brief  way  of 
asserting:   *No  entity  {x)  is  identical  with  any  entity 
other  than  what  is  identical  with  x!    This  axiom  ex- 
presses a  universal  and  rationally  grounded  truth,  ex- 
pressed in  terms  of  the  two   relations,   identity  and 
otherness.    I  shall  attempt  to  show  that  the  conceptions 
of  identity  and  of  otherness  are  two  independent  inde- 
finables^  the  understanding  of  which  is  required  in  order 
intelligently  to  accept  the  truth  of  the  above  axiom  or 
of  any  other  proposition  which  explicidy  or  implicidy 

13—2 


c|.  /Kc*^    '■Tm.J:^"-   C-^-^V  ^.    M-i.   P.  fsO 


196 


CHAPTER  XII 


THE  RELATION  OF  IDENTITY 


197 


involves  the  notions  of  identity  or  of  otherness.    Now 
the  above  axiom  of  identity  (xix)  is  never  explicitly 
used  anywhere  but  in  an  abstract  logical  or  philosophi- 
cal context;  on  the  other  hand  xiy  is  explicitly  used  in 
concrete  logical    and  mathematical  formulae;    and  its 
usage  in   such  cases  always  involves  the    process   of 
substituting y  for  x    The  connection  between  identity 
and  substitution  is  roughly  expressed  in  the  rule :  Given 
xiy,  we  may  always  substitute  y  for  x.     More  exactly, 
taking/  to  be  any  predication,  'If  ^  is  identical  with  j^, 
then  x\^p  would  imply  y  is  /.'    The  converse  of  this 
is  also  generally  admitted:  viz.  that  ^ If  for  every  value 
of/,  ^  is  /  would  imply  y  is/,  then  x  must  be  identical 
withjK.'    From  these  two  assertions  conjoined  it  follows 
that  the  proposition  'x  is  identical  with  j^/'  is  equipollent 
or  co-im'plicant  with  the  proposition  that  '  For  every 
predication  p,x\sp  would  imply  yisp:    The  problem 
thus  arises  whether  this  equipoUence  or  co-implication 
can  serve  as  a  definition  of  identity.     My  ground  for 
rejecting  such  a  view  is  that  the  equipoUence  asserted 
could  not  be  understood  or  utilised  unless  we  under- 
stood what  is  meant  by  the  assertions  'x  is  identical 
with  X,'  y  is  identical  withj)/'  and  '/is  identical  with/,' 
and  had  accepted  these  assertions  as  true,  because  of 
our  independent  understanding  of  what  is  meant  by  the 

relation  of  identity. 

In  the  converse  form  of  the  identity  formula  we  have 
admitted  that  if  all  the  adjectives  that  characterise  a 
substantive  x  also  characterise  a  substantive  y,  then  x 
and  y  are  identical.  But  the  hypothesis  here  is  really 
impossible,  for  the  adjective  'other  than  y'  cannot 
characterise  y.     Hence  there  is  one  adjective  at  least 


that  must  characterise  x  but  not  y.  What  then  is  the 
one  relational  adjective  that  must  characterise  any  one 
existent  and  not  any  other?  It  must  be  existence. itself. 
For  to  exist  means  to  stand  out  from  amongst  other 
things.  Otherne^  is  thus  presupposed  by  existence.  In 
short,  if  the  existent  jF  what  is  manifested  in  time  and 
space,  and  if  time  and  space  are  wholes  divisible  into 
parts,  then  the  only  necessarily  differentiating  mark  of 
one  existent  must  be  its  temporal  and  spatial  position. 
This  brings  us  back  to  the  Leibnizian  formula,  but  at 
the  expense  of  admitting  the  reality  of  time  and  space 
as  the  condition  of  otherness— a  condition  which  is  both 
necessary  and  sufficient. 

Although  in  the  sense  explained  identity  always 
implies  the  legitimacy  of  substitution,  we  cannot  say 
conversely  that  the  legitimacy  of  substitution  always 
implies  identity.  For  whenever  two  predications  are ' 
co-implicative,  the  one  may  always  be  substituted  for 
the  other  in  the  same  way  as  for  substantives  which  are 
identical.  Thus,  for  substantives  x  and  y,  we  have  the 
formula : 

If  x  is  identical  with  jjr,  then  'x  is/'  is  co-implica- 
tive  with  'y  is/,'  where/  is  any  predication  appli- 
cable to  x  and  y. 
Corresponding  to  this,  for  predications  q  and  r,  we  have: 
If  q  is  co-implicative  with  r,  then  'q  is  n  \%  co- 
implicative  with  'r  is  n,  where  n  is  any  predication 
applicable  to  q  and  r. 

Thus,  given  the  co-implication  of  the  two  predications 
{q)  human  being,  and  (r)  featherless  biped,  we  can  infer 
that  the  proposition  'the  number  of  human  beings  is  n 


A-m  "**'*^ 


Existence. 


r'A*"*' 


■mx^ 


k^     lUA 


— <c  4  i 


r»~«r 


♦  •*► 


ur* 


A-?"- 


^  / . 


kf-t 


198 


CHAPTER  XII 


THE  RELATION  OF  IDENTITY 


199 


is  co-impHcative  with  the  proposition  'the  number  of 
featherless  bipeds  is  n!  And  again  given  the  co-impli- 
cation of  the  two  propositions  {g)  'There  is  a  righteous 
God'  and  (r)  'The  wicked  will  be  punished,'  we  can 
infer  that  the  proposition  'That  there  is  a  righteous  God 
is  problematic'  is  co-implicative  with  the  proposition 
'That  the  wicked  will  be  punished  is  problematic'  These 
examples  show  that  the  relation  of  co-implication  be- 
tween predications  has  some  of  the  same  properties  as 
that  of  identification  between  substantives,  and  therefore 
co-implication  is  apt  to  be  conceived  (and  even  by  some 
symbolic  logicians  has  actually  been  symbolised)  as 
equivalent  to  identification.  It  appears  to  me  that  it  is 
theoretically  possible  that  the  conception  of  co-implica- 
tion could  be  shown  to  correspond  to  factual  identifica- 
tion ;  but  this  indeed  is  doubtful,  because  the  relation  of 
co-implication  is  compound,  i.e.  it  denotes  the  conjunc- 
tion of  the  two  correlatives  implying  and  implied  by, 
whereas  it  seems  impossible  to  reduce  the  notion  of 
identification  to  the  conjunction  of  two  correlatives. 

Before  dismissing  this  subject,  it  must  be  admitted 
that  both  as  regards  substitution  for  identified  substan- 
tive terms,  and  substitution  for  co-implicative  predica- 
tions, certain  limitations  seem  to  be  required.  For 
example  Mr  Russell  has  familiarised  us  with  illustra- 
tions for  the  necessity  of  this  limitation  by  such  ex- 
amples as  that,  from  the  identification  of  'Scott'  with 
'the  author  of  Waverley,'  we  cannot,  by  substituting  the 
one  term  for  the  other  in  such  a  proposition  as  'George  IV 
believed  Scott  to  have  written  Marmion  infer  that 
'George  IV  believed  the  author  of  Waver  ley  to  have 
written  Marmion'    Or  again  that  the  number  of  the 


apostles  is  identical  with  the  number  of  months  in  the 
year,  does  not  necessarily  imply  that  anyone  who  doubts 
that  the  number  of  apostles  is   12,  would  necessarily 
doubt  that  the  number  of  months  was  12.     It  appears 
that  the  only  statements  for  which  such  substitutions 
are  ijivalid  are  secondary  propositions  predicating  of  a 
primary  proposition  some  or  other  relation  to  a  thinker, 
v/    §  8.    To  complete  the  account  of  identity,  it  is  neces- 
sary to  anticipate  what  will  be  elaborated  in  a  later  part 
of  the  work;  and  for  this  purpose  the  discussion  will  be 
restricted  to  identity  of  substantives  proper,  excluding 
any  further  reference  to  the  identity  of  adjectives  or 
predications.    A  substantive  proper  I  have  djefined  as  . 
what  is  manifested  in  space  and  time,  or  otherwise  an  | 
existent.;  and  the  category  existent  has  been  divided  ! 
into  the  two  sub-categories  called  respectively  occurrent 
and  continuant.     Now  identity  as  applied  to  an  occur- 
rent could  be  illustrated  thus:  "The  flash  of  lightning 
to  which  I  am  pointing  is  identical  with  the  flash  of 
lightning  to  which  you  are  pointing.'    A  continuant,  on 
the  other  hand,  means  that  which  continues  to  exist 
while  its  states  or  relations  may  be  changing ;  identity 
"  of  continuant  may  therefore  be  illustrated  by  some  such 
examples  as  '  The  body  which  illuminates  the  earth  is 
identical  with  the  body  that  warms  the  earth';  'The 
person  who  was  experiencing  the  tooth-ache  is  identical 
with  the  person  who  intends  to  go  to  the  dentist.'   This 
last  conception  we  find  discussed  at  some  length  under 
the  heading  'Identity'  by  the  earlier  English  writers 
Locke,  Hume  and  Reid,  who  used  the  term  to  signify 
personal  identity  instead  of  giving  to  it  the  merely  rela- 
tional significance  of  the  more  modern  conception.    In 


200 


CHAPTER  XII 


THE  RELATION  OF  IDENTITY 


201 


<35^«***^ 


a. 


/^ 


i 


i. 


;. 


effect,  for  them  the  assertion  or  denial  of  identity  is 
equivalent  to  the  assertion  or  denial  that  there  is  a 
person  who  continues  to  exist  throughout  a  period  of 
time  in  which  his  various  experiences  may  be  altering 
in  character.  The  destructive  view  represented  by  H  ume 
regarded  experiences  as  what  I  call  occurrents,  any  one 
of  which  is  merely  replaced  by  another  in  the  course  of 
f  time.  This  destructive  view  is  equivalent  to  the  denial 
i  of  any  psychical  continuant.  Necessarily  an  occurrent 
experience  is  as  such  identical  with  itself;  what  Hume 
denied  was  that  kind  of  connection  between  one  such 
occurrent  and  another  which  constitutes  them  into  alter- 
able  states^ of  one  individual  self.  Hence  it  was  not  the 
general  conception  of  identity,  but  the  reality  of  the 
psychical  continuant  that  Hume  denied  and  that  Reid 
maintained.  When  this  old  problem  is  revived  at  the 
present  day,  it  is  usually  formulated  as  the  question 
whether  there  is  any  ultimate  philosophical  justification 
for  regarding  one  and  another  experience  as  belonging 
to  the  same  self  This  phraseology,  however,  is  mis- 
leading ;  for  it  appears  to  assume  the  existence  of  a  self, 
and  to  raise  only  the  question  as  to  whether  we  can 
refer  different  experiences  to  the  same  self;  whereas 
the  real  problem  is  whether  there  is  a  self  at  all.  In 
discussions,  by  Hume  and  others,  connected  with  that 
of  'personal  identity,'  we  find  the  same  problem  raised  as 
to  the  validity  of  the  more  general  notion  *  substance' — 
i.e.  in  my  terminology,  a  continuant  (whether  psychical 
or  physical).  Now  we  might  interpret  Kant  as  admit- 
ting the  validity  of  the  conception  of  a  physical  con- 
tinuant while  denying  that  of  a  psychical  continuant. 
On  the  other  hand,  Berkeley  more  obviously  supported 


the  notion  of  a  psychical  continuant  and  rejected  that 
of  a  physical  continuant.  At  the  present  day,  most 
philosophers  who  reject  these  notions  regard  them  as 
reached  by  a  constructive  process,  and  therefore  as 
being  merely  convenient  fictions.  To  assume  that  a 
notion  is  necessarily  fictitious  because  it  owes  its  origin 
to  a  constructive  process  is  fallacious,  especially  for  those 
who  accept  the  central  Kantian  position  according  to 
which  the  objective  validity  of  any  conception  is  estab- 
lished by  showing  it  to  be  the  result  of  a  synthetic, 
i.e.  constructive  act  of  pure  thought.  The  same  relation 
between  identity  and  the  notion  of  a  continuant  applies 
to  the  physical  as  to  the  psychical  continuant.  In  the 
physical  sphere,  those  who  reject  the  physical  continu- 
ant maintain  that  any  physical  event  is  a  mere  occur- 
rent which  is  replaced  by  another  occurrent;  and  that 
a  so-called  change  is  merely  the  temporal  succession  of 
one  and  another  occurrent.  Those,  on  the  other  hand, 
who  accept  the  notion  of  a  physical  continuant  maintain 
the  validity  of  the  notion  of  change  as  distinct  from 
mere  alternation,  and  as  therefore  presupposing  the  con- 
ception of  a  physical  continuant. 

We  may  indicate  more  positively  the  distinction 
between  the  two  views  which  respectively  reject  and 
accept  the  notion  of  a  continuant,  while  agreeing  in  the 
application  of  the  relation  of  identity.  Those  who  deny 
continuance  of  existence  as  well  as  those  who  affirm  it 
can  legitimately  collect  all  the  experiences,  occurring 
during  some  part  or  the  whole  of  what  we  call  an  indi- 
vidual's total  experience,  to  constitute  a  class  and  assert 
that  this  collection  is  identical  with  itself  On  the  other 
hand,  for  those  who  affirm  the  continuant,  the  collection  is 


0^ 


^ 


202 


CHAPTER  XII 


203 


« 


not  a  mere  plurality  but  a  specific  kind  of  unity ;  in  other 
words,  they  hold  that  an  intimate  bond  of  causality 
subsists  between  the  experiences  attributable  to  one 
individual  of  a  kind  which  does  not  subsist  between 
experiences  arbitrarily  selected  from  the  histories  of 
different  individuals.  The  notion  of  this  unique  kind 
of  bond  is,  on  this  view,  the  product  of  a  constructive 
act,  but  not  to  be  dismissed  on  this  ground  as  merely 
fictitious. 


CHAPTER  XIII 

RELATIONS  OR  TRANSITIVE  ADJECTIVES 

§  I.    So  far  we  have  treated  the  adjective  solely  in 
its  reference  to  the  substantive  which  it  characterises. 
We  have  now  to  consider  a  type  of  adjective  whose 
meaning  when  analysed  exhibits  a  reference  to  some 
substantive  other  than  that  which  it  characterises.  Thus 
we  may  characterise  a  certain  child  by  the  adjective 
'  liking  a  certain  book,'  or  a  certain  book  by  the  adjec- 
tive *  pleasing  a  certain  child.'    These  adjectives  predi- 
cated respectively  of  the  child  and  of  the  book;  are 
complex;  and  when  we  take  the  substantival  reference 
out  of  this  complex,  there  remains  the  term  'liking'  or 
'pleasing.'    Such  terms  do  not  function  as  completed; 
adjectives,    and   will    be    called   relational   adjectives./ 
Propositions  involving  adjectives  of  this  type  may  be 
ranged  in  a  series  according  to  the  number  of  substan- 
tives to  which  they  refer.    Thus,  in  the  following  ex- 
amples :  'A  is  wise,'  'A  likes  B;  'A  gives  X  to  B' 
'A  accuses  B  at  time  T oi  C  the  number  of  substan- 
tival  references  are  respectively  one,   two,   three  and 
four,  and  the  corresponding  adjectives  or  propositions 
may  be  called  monadic,  diadic,  triadic  and  tetradic. 

Taking  first  the  two-termed  relation,  let  us  con- 
sider the  proposition  'X  likes  F'  or  *^  is  greater 
than  Y:  The  notion  of  '  J^  as  liking  K'  or  of  'X  as 
being  greater  than  F'  is  to  be  distinguished  from  the 
notion  of  '  F  as  liking  X '  or  *  F  as  being  greater  than 


S^  iW.  §«. 


204 


CHAPTER  XIII 


X!  At  the  same  time  the  thought  of  any  assigned 
relation  of  ^  to  Y  involves  the  thought  of  a  definitely 
assignable  relation  oi  Y  X.o  X  \  for  example,  the  thought 
of  X  as  liking  Y  involves  the  thought  of  Y  as  pleas- 
ing X\  and  X  as  greater  than  Y  involves  Y  as  being 
less  than  X,  Two  relational  adjectives  such  as  liking 
and  pleasing,  or  greater  than  and  less  than — each  of 
which  in  this  way  involves  the  other — are  called  cor- 
relatives, and  either  one  is  said  to  be  the  converse  of 
the  other.  When  the  relation  is  expressed  by  a  transi- 
tive verb,  the  opposition  between  active  and  passive 
expresses  the  mutual  implication  of  correlatives  :  thus, 
'X  likes  Y'  means  '  Y  is  liked  by  ^,'  or  ^  F  pleases  X' 
means  'X  is  pleased  by  K'^  Except  in  the  case  of 
the  active  and  passive  voice,  there  is  no  general  rule 
of  language  according  to  which  the  converse  of  a  given 
relative  can  be  expressed,  and  therefore  a  special  know- 
ledge of  words  in  current  use  is  required  in  order  to  be 
able  to  express  a  relation  in  its  converse  form  ;  as 
when  we  pass  from  'X  is  greater  than  Y'  to  '  Y  is  less 
than  X;  or  from  'X  likes  Y'  to  *  F  pleases  X'  How- 
ever, the  fact  expressed  in  terms  of  a  relative  is  the 
same  as  the  fact  expressed  in  terms  of  its  converse, 
whether  the  terms  employed  are  philologically  cognate 
or  not. 

It  must  be  pointed  out  that  Miking  F'  or  Miking 
someone,*  etc.,  is  a  completed  adjective ;  and,  in  general, 
out  of  a  relational  adjective  we  may  construct  a  com- 

'  Comparing  'x  sleeps'  or  'x  is  sleeping'  with  'x  hits  j^'  or  *:x:  is 
hitting  J,'  and  noting  that  *•  sleeps'  is  an  intransitive  and  hits  a  transitive 
verb^  we  ought  properly  to  call  sleeping  an  intransitive  and  hitting  a 
transitive  adjective.  Thus  a  relation  is  properly  defined  as  a  transitive 
adjective^'  the  ordinary  adjective  being  distinguished  as  intransitive. 


RELATIONS  OR  TRANSITIVE  ADJECTIVES       205 

pleted  adjective  by  supplementing  the  substantival 
reference.  And  conversely  most  ordinary  adjectives  in 
use  can  be  analysed  so  as  to  elicit  a  relational  element 
as  a  component.  For  instance  'amiable'  contains  the 
relational  element  'liked  by,'  and  may  be  roughly  defined 
'liked  by  most  people.'  Again,  substantive  words  are 
constructed  out  of  relational  adjectives,  e.g.  'a  shep- 
herd' which  means  'a  person  who  takes  care  of  sheep.' 
It  is  noteworthy,  however,  that  to  take  the  substantives 
'shepherd'  and  'sheep'  as  examples  of  correlatives 
involves  a  double  error,  since  the  true  correlatives 
involved  in  the  meaning  of  shepherd  are  '  taking  care 
of  and  '  taken  care  of  by,'  which  are  adjectival  and  not 
substantival ;  while  the  meaning  of  the  word  '  sheep ' 
contains  no  relational  element  at  all. 

§  2.  Our  immediate  concern  will  be  with  diadic 
adjectives,  otherwise  called  coupling^.  Given  any  two 
substantives — say  x  and  y — we  may  construct  what  will 
be  termed  a  substantive-couple  expressed  by  the  phrase 
^  X  tojv,'  which  is  to  be  distinguished  from  the  substan- 
tive couple  *jK  to  x!  Similarly,  given  any  two  correlative 
coupling  adjectives — say  greater-than  and  less-than — 
we  construct  what  will  be  termed  an  adjective-couple, 
expressed  by  the  phrase  'greater-than  to  less-than.' 
The  significance  of  a  substantive-couple  is  to  be  ex- 
plained by  defining  it  as  that  which  may  be  characterised 
by  an  adjective-couple ;  and  the  significance  of  an 
adjective-couple,  by  defining  it  as  that  which  may 
characterise  a  substantive-couple.  Thus  the  relation  of  | 
substantive-couple  to  adjective-couple  is  the  same  as 
that  of  an  ordinary  adjective  to  an  ordinary  substan- 
tive ;   and  just  as  the  latter  are  united  through  the 


206 


CHAPTER  XIII 


characterising  tie,  so  are  the  former.  Again,  just  as 
the  extension  determined  by  an  ordinary  adjective 
comprises  the  substantives  of  which  the  adjective  may 
be  truly  predicated,  so  we  may  say  of  an  adjective- 
couple  that  the  extension  which  it  determines  comprises 
the  substantive-couples  of  which  the  adjective-couple 
may  be  truly  predicated.  This  relation  between  the 
substantive-couple  and  the  adjective-couple  is  brought 
out  by  expressing  the  proposition  'x  is  greater  than^' 
in  the  form 

'x  to y  is-as  greater  than  to  less  than,' 

and  the  proposition  'y  is  less  than  x'  in  the  form 
'y  to  X  is-as  less  than  to  greater  than.' 

This  mode  of  formulation  helps  perhaps  to  explain  the 
process  of  relational  conversion,  which  may  be  illus- 
trated as  follows : 

( I )  :r  is  greater  than  y^ 

,\  {2)  X  to  y  is-as  greater  than  to  less  than, 

.'.  (3)^  to  X  is-as  less  than  to  greater  than, 

.  • .  (4)  jy  is  less  than  x. 

In  passing  from  (i)  to  (2),  the  introduction  of  the 
term  *  less  than '  depends  merely  upon  knowledge  of 
the  arbitrary  usage  of  language  ;  but  the  logical  validity 
of  the  step  rests  upon  the  fundamental  principle  of 
thought  that  every  relation  has  its  converse.  Each  step 
also  requires  that  the  order  in  which  the  adjective 
terms  are  mentioned  is  to  be  understood  to  correspond 
to  that  in  which  the  substantive  terms  are  mentioned. 
Similar  reformulations  could  be  applied  to  triadic  and 


RELATIONS  OR  TRANSITIVE  ADJECTIVES       207 

higher  orders  of  adjectives  :  thus  'x  receives  d  (romy 
will  be  rendered 

x\b\y  I  is-as  |  receiving :  given  by  :  giving  to ; 

and  'y  gives  ^  to  ;t: '  becomes 
y  \b  \x  I  is-as  |  giving  :  given  to  :  receiving  from  ; 

where  ' : '  stands  for  *  to,'  and  the  ordering  of  the  words 
is  to  be  interpreted  cyclically. 

§  3.  The  general  notion  of  an  adjective-couple  that 
can  be  predicated  of  a  substantive-couple  is  familiarly 
illustrated  in  what  is  called  analogy.  Take  the  proposi- 
tion *  England  to  Australia  is-as  parent  to  child  ' ;  here 
the  predicate  is  what  I  call  an  adjective-couple,  consti- 
tuted by  taking  the  relation  *  parent-of '  and  its  converse 
*child-of';  while  the  subject  is  a  substantive-couple 
composed  of  the  two  substantives  England  and  Aus- 
tralia. The  copula  '  is-as '  marks  a  statement  of  analogy. 
Another  example  with  the  same  adjective-couple  is 
*  France  to  Algiers  is-as  parent  to  child.'  From  these 
two  predications  of  the  same  adjective-couple  we  should 
infer  that  *  France  is  to  Algiers  as  England  is  to 
Australia.'  This  form  of  proposition,  however,  differs 
importantly  from  that  which  predicates  an  adjective- 
couple  of  a  substantive-couple.  For  in  affirming  the 
equivalence  of  the  relation  in  which  France  stands  to 
Algiers  with  that  in  which  England  stands  to  Australia, 
there  is  no  indication  of  the  kind  of  relation  in  respect 
of  which  the  two  substantive-couples  agree.  We  might, 
in  fact,  compare  this  inference  with  the  inference  that 
'X  is  like  F'  which  could  be  drawn  from  the  two 
propositions  that  'X  is  red  '  and  '  F  is  red ';  or  equally 
from  the  two  propositions  that  'X  is  square '  and  '  Y  is 


208 


CHAPTER  XIII 


square.'    We  then  see  that  such  terms  as  similar  or 
analogous  when  used  to  connect  two  substantives  or 
two  substantive-couples  are  quite  indeterminate  with 
respect  to  the  grgund  of  similarity  or  analogy  ;  so  that 
ifrom  the  assertions   'A  is  similar  to  B'  and  'B  is 
i similar  to  C  it  could  never  be  inferred  that  'A  is 
•  similar  to  C/    My  formulation  of  the  relational  proposi- 
tion 'A  to  B  is-as  /  to  ^ '  is  of  course  suggested  by  the 
arithmetical  expression  for  a  ratio  and  the  equality  of 
ratios,  but  here  there  is  a  danger  of  misunderstanding ; 
for  in  my  phraseology  we  could  assert  that '  8  to  6  is-as 
greater  by  2  to  less  by  2,'  and  again  that  *  5  to  3  is-as 
greater  by  2  to  less  by  2 '  and  hence  that  *  8  to  6  is-as 
5  to  3 ' :  but  in  arithmetic  the  phraseology  */  to  ^'  is 
understood  to  denote  exclusively  a  relation  under  the 
1  genus  called  ra^to,  and  could  not  be  applied  to  a  rela- 
tion  under  any  other  genus  such  as  difference  or  dis- 
tance, etc.,  etc.    Instead  of  calling  such  a  term  as  ratio, 
difference  or  distance,  etc.,  by  the  familiar  name^^;^^^^, 
it  ought,  properly  speaking,  to  be  termed  *  relational 
determinable'   in   contradistinction   to  three-fifths,  or 
minus  two,  or  three  yards  distant,  which  are  'relational 
determinates'  under  their  respective  relational  deter- 
minables.    Whereas  Aere  the  phraseology  '  a  to  d  is-as 
/  to  ^ '  is  used  for  relational  determinates  under  any 
relational  determinable,  in  arithmetic  this  phraseology 
is  limited  to  relational  determinates  under  the  one  re- 
lational determinable  called  ra^zo.    It  must  be  observed 
that,  when  the  term  analogy  is  explicitly  referred  to  a 
given  relational  determinable,  such  as  ralio,  then,  from 
the  assertion  that  two  substantive  couples  are  analogous 
to  the  same  substantive  couple,  we  may  infer  that  the 


RELATIONS  OR  TRANSITIVE  ADJECTIVES      309 

two  are  analogous  to  one  another.  This  is  precisely 
parallel  to  the  case  of  agreement  or  similarity  when 
explicitly  referred  to  any  given  adjectival  determinable, 
such  as  colour. 

§  4.  This  account  of  relational  adjectives  leads  to 
I  a  consideration  of  a  species  of  tie  distinct  from  the 
characterising  tie,  which  we  shall  call  the  cguj> ling  tie. 
In  the  phrase  '  x  to  y'  the  word  to  has  been  chosen  to 
indicate  this  tie,  and  hence  the  effect  of  the  coupling 
[tie  is  to  construct  a  substantive-couple.  Any  prepo- 
sition or  prepositional  phrase  such  as  of,  by,  for,  at^ 
\withy  in,  in  reference  to,  indicates  the  presence  of  the 
coupling  tie.  We  must  not,  however,  in  general  say 
that  the  preposition  denotes  merely  a  tie;  for  a  dif- 
ference of  preposition  often  stands  for  a  difference  in 
the  relation  predicated.  For  example  '  x  is  influenced 
to  move  towards  y'  has  a  different  meaning  from  '  x  is 
influenced  to  move  away  from  y' ;  where  the  difference 
of  preposition  is  seen  to  entail  a  difference  of  relation — 
namely  the  difference  between  attraction  and  repulsion. 
In  fact,  prepositions  used  along  with  adjectives  or  verbs 
[express    determinate   modifications   of    relation.     The 

essential  feature  of  a  tie,  on  the  other  hand,  is  that  it  is 
[incapable   of  modification,   and   hence  we  frequently 

md  that  it  does  not  enter  as  a  separate  verbal  com- 

)onent  in  a  sentence. 

Whenever  a  tie  (whether  it  be  the  characterising 
tie,  or  the  coupling  tie,  or  any  other)  does  not  appear  as 

in  actual  word,  there  are  conventions  of  language  which 

indicate  its  presence.    In  languages  in  which  inflexion 

[is  largely  used,  such  as  Latin  and  German,  there  are 

:wo  main  kinds  of  grammatical  rule ;  namely,  the  rules 


J.  L. 


14 


210 


CHAPTER  XIII 


of  concordance  and  the  rules  of  governance.    We  shall 
find  that  the  rules  of  concordance  point  to  the  presence 
of  the  characterising  tie  ;  and  those  of  governance  to 
the  coupling  tie.    The  rules  of  concordance  are,  briefly, 
that  adjectives  and  verbs  must  agree  in  gender,  number 
and  case,  with  the  substantives  that  they  characterise ; 
so  that  the  characterising  tie  is  not  necessarily  expressed 
by  use  of  the  verb  *  to  be '  but  merely  by  inflexion.  On 
the  other  hand,  the  rules  of  governance  always  deter- 
mine the  case — genitive,  dative,  accusative,  or  ablative 
— of  the  substantive  that  is  introduced  along  with  any 
transitive  verb,  relational  adjective,  or  preposition.  We 
find,  especially  in  Latin,  that  considerable  changes  in 
the  order  of  words  (which  may  vary  for  purposes  of 
rhetorical  significance)  are  permissible  because  of  the 
inflexions  which  are  understood  to  indicate  (i)  by  gram- 
matical agreement,  how  the  words  are  to  be  attached 
in  thought  by  the  characterising  tie,  and  (ii)  by  gram- 
matical governance,  how  they  are  to  be  attached  in 
thought   by  the    coupling    tie.     Furthermore,    where 
modification  of  case  occurs  (with  or  without  a  prepo- 
sition), not  only  the  coupling  tie,  but  also  the  special 
modification  of  relation  that  is  to  be  understood   is 
grammatically  indicated.  The  characteristic  of  English, 
in  contrast  to  highly  inflexional  languages,  is  that  no 
inflexions  are  required  by  rules  either  of  concordance 
or  of  governance,  except  in  the  two  instances:  (i)  fori 
differences  of  person  and  number  in  many  verbs  (which 
illustrate  the  characterising  \\€),  and  (ii)  the  accusatives  I 
— him,  her,  me,  us,  them,  whom  (which  illustrate  the 
coupling  tie).    All  the  other  instances  of  inflexion  in 
English — for  example  the  possessive  pronouns  and  the 


RELATIONS  OR  TRANSITIVE  ADJECTIVES       2ir 

lenses  of  verbs — are  used,  not  according  to  any  rules 
)f  concordance  or  governance,  but  to  express  distinc- 
;ions  of  meaning.  The  difference  between  the  two 
:inds  of  inflexion — the  one  being  significant  and  the 
)ther  syntactic — is  brought  out  by  comparing  the 
English  *her  father'  or  *  his  mother,*  where  the  differ- 
ence of  gender  is  significant  and  not  syntactic,  with  the 
rench  '  son  pere '  or  '  sa  mere,'  where  the  difference 
)f  gender  is  syntactic  and  not  significant.  In  English, 
|he  conventional  rules  of  concordance  or  governance 
ire  replaced — except  in  the  two  cases  mentioned  above  i 
-by  the  equally  conventional  ordering  of  the  words.     \ 

§5.  The  coupling  tie — which  might  have  been  called 

le  prepositional  tie,  in  consideration  of  the  grammati- 

lal  rules  of  governance,  or  again  the  relational  tie,  in 

pnsideration  of  the  philosophical  problems  that  have 

jeen  raised  in  regard  to  the  nature  of  relation — is  of 

mdamental  importance  in  discussing  one  of  the  para- 

loxes  that  Mr  Bradley  and  others  have  found  in  the 

eneral  notion  of  relation.  The  paradox  is  briefly  brought 

ut  in  the  following  contention:  when  we  think  oi x  as 

eing  r  to  y,  we  have  first  to  relate  x  to  yhy  the  rela- 

[on  r,  and  then  relate  the  relation  r  \.o  x  by — say  r' — 

A  r  \.o  y  by — say  r^',  another  relation.    This  again 

^ill  require  that  x  should  be  related  to  /  by  the  further 

dation  /",  which  will  lead  to  an  infinite  regress  on  the 

|de  of  X,  and  a  similar  regress  on  the  side  of  y.    This  | 

iradoxical  contention  is  met  by  pointing  out  that  in  1 

>nstructing  an  object  out  of  the  constituents  x,  r,  and  j, 

[e  do  not  introduce  another  constituent  by  the  mere 

:t  of  constituting  these  constituents  into  a  unity.    The  j 

•etence  of  paradox  is  due  to  the  assumption  that  to ' 

14 — 2 


u.>^^»*^*»^*< 


l/U 


-^Lw^-vwi/M^. 


212 


CHAPTER  XIII 


i  the  act  of  relating  or  constructing  there  corresponds 
a  special  mode  of  relation ;  so  that  a  tie  is  confused  with 
a  relation.  That  a  tie  and  a  relation  are  distinct  is 
brought  out  by  considering  the  fact  that  if,  for  a  given 
adjective — whether  ordinary  or  relational — we  substi- 
tute another  adjective,  we  shall  have  constructed  a 
different  unity;  but,  if  we  drop  the  characterising  tie 
with  a  view  to  replacing  it  by  some  adjective  or  rela- 
tion, then  either  t_he_  unky  itself  is  destroyed,  or  it  will 
be  found  that  the  characterising  tie  remains  along  with 
the  adjective  or  relation  so  introduced.  Similarly,  the 
coupling  of  terms  is  not  a  mode  of  relating  them  for 
which  another  mode  of  relation  could  be  substituted ; 
for,  if  they  were  uncoupled,  again  the  unity  would  be 
destroyed. 

The  distinction  between  a  tie  and  a  relation  may  be 
brought  out  from  another  point  of  view  by  the  con- 
sideration that  the  specific  difference  between  one  kind 
of  tie  and  another  is  determined  by  the  logical  nature 
'  of  the  constituents  tied.  Thus  the  use  of  an  adjective 
in  general  involves  the  characterising  tie,  by  which  it 
is  attached  to  a  substantive ;  and  the  use  of  a  relational 
adjective  in  particular  further  involves  the  coupling  tie 
by  which  the  two  substantive-terms  are  attached  to  one 
another.  On  the  other  hand,  where  terms  are  related 
by  a  genuine  relation,  their  logical  nature  allows  any 
specific  relation  to  be  replaced  by  any  other,  this  other 
being  in  general  under  the  same  relational  determinable. 
Now  just  as  we  must  distinguish  between  any  rela- 
tion and  the  relational  (or  coupling)  tie,  so  I  have 
throughout  assumed,  and  it  is  of  the  first  importance  to| 
emphasise,  the  distinction  between  characterisation  and 


RELATIONS  OR  TRANSITIVE  ADJECTIVES       213 

the  characterising  tie.    I  have,  in  fact,  spoken  repeatedly 
of  the  relation  of  adjective  to  substantive;  and  this  is 
the  relation  called  characterisation — a  specific  kind  of 
relation  to  be  distinguished,  for  instance,  from  *  liking' 
or  'exceeding,'  etc.    While  characterisation,  then,   is 
a  relation,  the  characterising  tie  (like  any  other  tie)  is 
a  mode  of  connection,  represented  usually  by  the  par- 
ticiple 'being.'    This  participle  may  be  expanded  into 
*being-characterised-as-being,'and  it  will  still  represent 
merely  the  characterising  tie.    Thus  the  following  series      - 
of  examples  are  seen  to  be  but  different  modes  of  ex-     ^^  ^^^^^^^  ^ 
pressing  precisely  the  same  fact :   I  am  thinking  ( i )  of    ^^^  j  ju^ 
this  '  as  I  tall;  (2)  of  this  [as-being  |  tall;  (3)  of  this  |  as-       ,^ 
being- characterised-as -being  i  tall;  where  the  phrase 
enclosed  in  vertical  lines  represents  nothing  more  nor 
less  than  the  characterising  tie.    The  equivalence  be- 
tween *5  as-being  P'  and  'S  as-being  characterised  as- 
being  /*,'  etc.,  is  precisely  analogous  to  the  arithmetical 
equalities:   '5x/^*  = '5x  i  x/^' =  *6'x  i  x  i  x  P,'  where 
5  stands  for  any  magnitude  and  P  for  any  numerical 
multiplier,  while  the  number  one  takes  the  place  of  the 
relation    'characterised^   and   the   operator  '  x '   takes 
the  place  of  the  tie  'as-being.'    We  will  therefore  call 
'characterisation'  the  unit  relation,  because  it  may  be     ^ 
conceived  as  a  factor  in  every  adjective  and  in  every  re-     " 
lation.    The  conception  of  '  characterised '  as  a  relational 
factor,  analogous  to  one  in  the  expression  5x  (i  xP), 
will  be  shown  more  precisely  by  adopting  a  different 
mode  of  bracketing  (3)  whereby  it  becomes:  (4)  I  am 
thinking  of  'this  |as-being|  characterised-as-being-tall.' 
Now  this  last  expression  is  formally  analogous  to(5)  I  am 
thinking  of  this  |  as-being  1  taller-than-that.   The  phrase 


^lAfJ^.. 


^     A. 


Wr^^ 


-< 


214 


CHAPTER  XIII 


I  as-being  |,  occurring  both  in  (4)  and  (5),  represents  the 
characterising  tie ;  but  in  (5)  the  relation  (with  its  tie) 
is  expressed  by  *taller-than,'  and  supplemented  by  the 
substantive-term  *that,'  while  in  (4)  the  relation  (with 
its  tie)  is  expressed  by  'characterised-as-being'  and  sup- 
plemented by  the  adjective-term 'tall'  Since,  moreover, 
we  may  expand  the  tie  in  (5)  into  'as-being-character- 
ised-as-being,'  we  see  that  'characterised'  is  a  latent 
factor  in  every  relation,  as  well  as  in  every  ordinary 
adjective.  Thus  *  characterised'  is  a  latent  factor  even 
in  the  relation  *  characterising':  for  'I  am  thinking  of  P 
as  characterising  5'  may  be  expanded  into  *  I  am  think- 
ing of  P  j  as-being  |  characterised  as  characterising  S! 

j  This  reformulation  incidentally  shows  that,  though  char- 
acterised has  the  properties  of  a  unit  relation,  yet  its 

'  converse  characterising  has  not  these  properties. 

§  6.  Now  the  form  of  proposition  in  which  'character- 
ised' is  introduced  explicitly  as  a  relation,  derives  its 
significance  and  its  legitimacy  from  our  having  taken  an 
adjective — namely  'tall' — as  a  term.  We  are  therefore 
extending  the  application  of  the  notion  of  a  relation, 
when  in  this  way  we  take  an  adjective  as  term^  instead 
of  (as  hitherto)  a  substantive.  Indeed  no  limit  can  be 
imposed  upon  the  kind  or  category  of  entity  which  may 
constitute  a  'term'  of  which  adjectives  or  relations  to 
other  entities  may  be  predicated.  In  particular,  we  must 
recognise  that  certain  adjectives  may  be  significantly 
predicated  of  adjectives  and  of  propositions,  and  even  of 
relations ;  and  that  certain  relations  may  be  significantly 
predicated  as  subsisting  between  a  substantive  and  an 
adjective,  or  between  one  adjective  and  another,  or 
between  one  proposition  and  another,  or  even  between 


RELATIONS  OR  TRANSITIVE  ADJECTIVES       215 

one  relation  and  another.  The  adjectives,  relations, 
or  propositions  of  which  other  adjectives  or  relations 
may  be  predicated  must  when  so  connected  be  called 
terms,  in  contrast  with  the  adjectives  or  relations  pre- 
dicated of  them.  The  logical  mode  in  which  adjectives, 
relations  or  propositions  enter  as  terms  into  a  construct, 
is  reflected  in  language  by  the  substantival  form  assumed 
by  them ;  e.g.  intolerance,  hatred,  the  enthusiasm  of  the 
people,  that  matter  exists,  to  be  or  not  to  be,  etc.  etc. 

Underlying  the  merely  nominal  question  whether 
adjectives,  relations  or  propositions  when  functioning 
as  substantives  in  a  construct  should  be  called  terms, 
two  philosophical  questions  arise,  which  I  shall  here 
deal  with  rather  summarily.     First,  is  it  literally  the   I, 
same  entity  which  can  be  treated  indifferently  either  as 
an  adjective  in  its  primary  or  natural  functioning,  or 
as  a  quasi-substantive  of  which  certain  other  adjectives 
or  relations  may  be  predicated?    To  this  I  give  an 
affirmative  answer;  and  the  objections  to  my  view  are 
(I  think)  met  by  insisting  that  the  adjectives  or  rela- 
tions which  may  be  significantly  predicated  oi  primary    yf^  ^ 
adjectives  or  relations  (as  they  may  here  be  called)     .f.^i^j^f^^ 
belong  to  a  different  logical  sub-category  from  these   %^;jf^^;^ 
latter,  and  may  be  called  secondary.    Thus,  it  is  not    ^^^  Id^^us* 
that  the  primary  adjective  changes  its  category  when   ^^"[^^ 
functioning  as  quasi-substantive,  but  it  is  that  the  second-   '"^^^  **"''^ 
ary  adjective  must  be  said  to  belong  to  a  special  sub- 
category, determined  by  the  category  of  the  primary 
adjective  of  which  it  may  be  predicated.    But  a  second 
question  arises:  whether  the  relation  or  adjective  that 
is  apparently  predicated  of  a  relation,  adjective  or  pro- 
position, is  really  ^o  predicated;  or  whether  it  is  pre- 


2l6 


CHAPTER  XIII 


dicated  rather  of  certain  substantives  to  which  there  is 
implicit  or  explicit  reference.  As  regards  this  question, 
it  must  be  pointed  out  that  while  relations,  subsisting 
primarily  between  certain  entities  entail  relations  be- 
tween other  entities  involved  in  or  connected  with  the 

I  former,  yet  the  relations  thus  entailed  are  not  identical 

I  with  the  primary  relation ;  so  that  whether  a  relation  is 
predicated  of  this  or  of  that  kind  of  entity  needs  separate 
discussion  in  each  type  of  case.  A  typical  example  will 
be  discussed  in  a  succeeding  paragraph. 

§  7.  We  now  propose  to  analyse  relational  proposi- 
tions of  more  complicated  forms.  The  principles  (pre- 
viously illustrated)  in  accordance  with  which  a  diadic 
proposition  may  be  reduced  to  a  monadic  by  hyphening 
(or  bracketing)  the  predicate,  and  may  be  converted 
by  substituting  for  the  given  relation  its  correlative, 
may  be  extended  to  relational  propositions  of  higher 
orders.  In  such  propositions  any  one  of  the  substantive- 
terms  may  be  taken  as  subject,  and  a  complex  containing 

'  the  remaining  substantive-terms  as  monadic  predicate; 
or  any  couple  of  substantive-terms  may  be  taken  as 
subject  and  a  complex  containing  the  remaining  sub- 

I  stantive-terms  as  diadic  predicate;  and  so  on.  In  such 
trsLnsformaitions, 3.ny permutations  thsLt  are  made  amongst 
the  substantive-terms,  will  require  the  substitution  of 
'cognate'  modes  of  expressing  the  given  relation;  and, 
as  in  diadic  relations,  the  order  of  relationality  is  reduced 
by  constructing  a  complex  predication  indicated  by  a 
hyphen  or  bracket.  For  example,  the  triadic  proposition 
*A  gives  ^  to  ^'  is  reduced  to  a  diadic  by  transforming  it 
into  'A  gives- A'  to  B'  or  'B  is  receiver-from-^  oi X'  or 
'X  is  given-to-^  by  Ay'  which  again  may  be  converted, 


RELATIONS  OR  TRANSITIVE  ADJECTIVES       217 

respectively,  into  'B  receives- A'  from  A,'  'X  is  given- 
by-^  to  B,'  'A  is  giver-to-i9  of  X'  Thus  the  single 
radical  relation  represented  by  such  a  verb  as  *to  give,* 
from  which  a  triadic  proposition  is  constructed,  gives 
rise  to  three  pairs  of  converse  forms  (making  six  corre- 
latives), namely :  giver-to  and  receiver-from ;  receiver- 
of  and  given-to;  given-by  and  giver-of  And,  further- 
more, each  of  the  above  six  diadics  becomes  monadic 
by  completing  the  hyphening  of  the  predicate  without 
any  further  verbal  alteration. 

In  the  above  illustrations  a  single  radical  relation 
was  involved;  but  another  kind  of  complication  arises 
when,  besides  a  number  of  substantival  components,  the 
proposition  contains  more  than  one  adjectival  or  rela- 
tional zom^ow^wx.,  these  being  in  general  indicated  by 
different  verbs.  For  example,  the  proposition  'A  pre- 
vented B  from  hurting  C  contains  the  three  substantive- 
terms  Ay  By  Cy  together  with  two  verbs,  of  which  'pre- 
vent' is  the  principal  and  'hurt'  is  sub-ordinate.  Such 
a  proposition,  treated  merely  as  triadic,  may  be  trans- 
formed by  permutation  into  (e.g.)  '^ was  prevented  by^ 
from  hurting  C,'  and  by  bracketing  into  *  C  was  saved- 
from-being-hurt-by-^  by  ^.'  But  another  form  of  ana- 
lysis may  be  used  where — as  in  the  case  before  us — 
we  are  dealing,  not  only  with  a  plurality  of  substantive 
terms,  but  also  with  a  plurality  of  radically  different 
relational  components.  In  this  analysis,  we  take  as  a 
bracketed  constituent  a  complex  containing  all- the  com- 
ponents of  a  complete  proposition ;  viz.  the  proposition 
that  contains  the  subordinate  relation  or  adjective.  A 
proposition  in  this  aspect  may  be  called  a  p±ssibile. 
Ih^  possibile  in  our  illustration  is  '^*s-hurtingC,'  and 


2l8 


CHAPTER  XIII 


the  proposition  predicates  of  the  person  A  the  relation 
*  preventing.'  The  distinction  between  the  two  modes 
of  bracketing — in  both  of  which  a  triadic  proposition  is 
reduced  to  a  diadic — will  be  shown  by  comparing:  ^A 
is  I  preventer-from-hurting-C|  by  B'  with  'A  |prevents| 
the-hurting-of-C-by-^.'  In  the  former  the  terms  of  the 
relation  are  the  two  persons  'A'  and  'By  and  the  rela- 
tion may  be  expressed : 

^A'  to  'B'  is-as  'preventer-of-hurt-to-C-from'   to 
*prevented-from-hurting-C  by'; 

in  the  latter  the  terms  are  the  person  'A^  and  the  pos- 
sibile  'Os  being  hurt  by  B,'  the  relation  being: 

'A'  to  'Os  being  hurt  by  B'  is-as  *preventer-of 
to  'prevented-by.* 

The  immediate  purpose  of  this  illustration  is  to  show 
that  a  proposition  that  predicates  a  relation  of  a  term 
to  a  possibile,  entails  also  the  predication  of  certain 
complex  relations  between  this  principal  term  and  the 
!  various  substantive  constituents  of  the  possibile.  The 
example  under  consideration  may  be  more  fully  ex- 
pressed by  explicit  reference  to  the  action  of  A  which 
was  causally  preventive:  thus,  *yi's-instructing-Z?  pre- 
vented ^  s  hurting- C/  This  more  explicit  form  of  state- 
ment is  typical  of  the  causal  proposition  as  such  which, 
properly  speaking,  always  relates  on^possibile  to  another 
either  by  way  of  production  or  of  prevention.  For  ex- 
ample, such  a  statement  as  *The  earth  causes  the  fall 
of  the  stone'  is  an  elliptical  expression  for  the  more 
fully  analysed  proposition  *The-proximity-of-the-earth 
causes  the-fall-of-the-stone.'  Since,  however  (as  follows 
from  the  above  remarks),  a  diadic  relation  of  one  pos- 


RELATIONS  OR  TRANSITIVE  ADJECTIVES       219 

sibile  to  another  entails  certain  more  complex  relations 
amongst  the  substantive  constituent^  of  the  possibilia, 
it  is  formally  legitimate  to  assert  that  the  earth  stands 
in  some  relation  to  the  fall  of  the  stone,  or  again  that 
the  earth  stands  in  some  relation  to  the  stone,  though 
each  of  these  relations  is  more  complex  than  the  diadic 
relation  of.  causation,  which  holds  between  the  two 
possibilia. 

§  8.  The  above  discussion  leads  to  the  special  pro- 
blem of  the  nature  of  the  relation  of  assertion.  Consider 
the  simplest  case :  'A  asserts  5  to  be  characterised  by 
P'  This  contains  the  three  terms  Ay  S  and  /*,  together 
with  the  two  verbs  or  relations  'assert'  and  'character- 
ise/ of  which  'assert'  is  the  principal,  and  'characterise' 
the  subordinate.  Taking  the  complex  as  expressing  a 
triadic  relation  oi  A  to  S  to  P'y  it  may  be  reformulated 
in  several  ways :  such  as,  *A  asserts-/*-to-characterise5'; 
'P  is  asserted-by-^-to-characterise  5,' which  predicate 
diadic  relations  of  A  to  S,  and  of  P  to  5  respectively. 
But  the  more  natural  mode  of  expressing  the  relation 
is  as  one  of  the  thinker  A  to  the  possibile  'S  being 
(characterised  by)  P,'  My  account  of  the  relation  of 
causation  as  holding  primarily  between  possibilia  but 
also  as  entailing  relations  (of  a  higher  order  than  diadic) 
amongst  the  component  terms  of  the  possibilia,  must  be 
applied  to  any  such  relation  as  that  of  interrogating, 
doubting,  considering,  affirming,  denying,  etc.,  in  which 
a  thinker  may  stand  to  a  possibile  or  assertibile.  The 
relation  of  the  thinker  to  the  proposition  as  a  whole 
does  not  preclude — it  rather  entails — relations  to  the 
constituents  of  the  proposition.  Thus,  the  relation  of  1 
the  thinker  to  the  subject  S  is  expressible  in  terms  oiP, 


220 


CHAPTER  XIII 


viz.,  *  asserts-/^. to-characterise' ;  and  that  of  the  thinker 
to  the  predicate  P  is  expressible  in  terms  of  S,  viz., 
*asserts-S-to-be-characterised-by.' 

This  principle— that  assertion  or  judgment  involves 
a  relation  of  the  thinker  to  each  of  the  constituents  of 
the  proposition  as  well  as  to  the  proposition  as  a  unitary 
whole — is  familiarly  (but  misleadingly)  expressed  in 
some  such  form  as  that  *in  judgment  the  thinker  asserts 
a  relation  of  one  idea  to  another.'    If  by  idea  was  really 
meant  'the  object  of  a  thought'  rather  than  *the  thought 
of  an  object,'  this  statement  would,   in  my  view,   be 
essentially  correct.    Or  more  briefly :  I  agree  that  judg- 
ment relates  the  object  of  one  thought  to  the  object  of 
another  thought;  but  I  deny  that  judgment  relates  the 
thought  of  one  object  to  the  thought  of  another.    That 
is  to  say,  though  judgment  requires  us  to  think  about 
objects,  the  judgment  \^about\ki^^^  ol)jects,and  not  about 
our  thinking  about  them.    The  account  of  judgment  that 
is  to  be  rejected  involves  a  confusion  between  a  primary 
proposition  which  is  about  objects,   and  a  secondary 
proposition  (as  it  may  here  be  termed)  about  our  ideas 
of  objects ;  and,  if  this  identification  of  a  primary  with 
a  secondary  proposition  is  consistently  carried  out,  we 
should  have  to  interpret  a  secondary  proposition  as  a 
tertiary ;  namely,  as  being  about  our  ideas  about  our  ideas 
of  objects,  and  so  ad  infinitum.    But  admitting,  as  I  have 
done,  that  in  judgment  we  assert  a  relation  of  one 
object  of  thought  to  another,  say  of  S  to  P,  it  is  neces- 
sary further  to  consider  what  sort  of  idea  we  have  of 
5  or  of /*  when  we  judge  that 'S  is  P' \  and  here  I  pro- 
pose to  distinguish  between  the  specific  and  the  generic 
idea  of  5  or  of  P.   The  specific  idea  of  S  that  is  involved 


RELATIONS  OR  TRANSITIVE  ADJECTIVES       221 

in  doubting  or  asserting  the  proposition,  is  the  idea  of  5as 
'characterised  by  P'\  and  this  idea  includes  the  ^^^^^r^i:  ^ 
idea  of  5  as  'characterisable,'  i.e.  as  of  the  nature  of  a 
substantive.  Similarly  the  specific  idea  of  P  is  the  idea 
oi  P  as  *  characterising  5"  and  this  includes  \!ci^  generic 
idea  of  P  as  'characterising ' ;  i.e.  as  of  the  nature  of  an 
adjective.  But  here  it  is  to  be  observed  that  the  relation 
that  may  be  said  to  be  predicated,  viz.  that  of  character- 
isation, does  not  subsist  between  the  idea  of  5  and  the 
idea  of  P,  since  each  of  these  ideas  is  specifically  com- 
pleted in  the  single  complex  idea  of  '5-as-characterised- 
by-/^'  or  of  'P-as-characterising-5"  or  again  of  'the- 
characterisation-of-5-by-/^';  and  I  hold  that  these  three 
phrases  express  different  modes  of  constructing  on^  and 
the  same  construct  or  complex  object  of  thought.  No 
difference  of  principle  is  involved  when  dealing  with  an 
explicitly  material  relational  proposition,  such  as  '5  is 
i?  to  T'  Here  the  specific  idea  of  5  is  the  completed 
idea  of  5  as  'being  i^  to  T'\  that  of  7^ is  the  completed 
idea  of  T  as  'being  7?  to  5";  that  oi  R,  as  'relating  5 
to  T'\  and  that  of  R  as  'relating  7^ to  S!  These  specific 
ideas  include  such  generic  ideas  as  that  of  S  and  of  T 
as  being  substantives  and  of  R  and  R  as  being  a 
pair  of  relations  correlative  to  one  another.  As  in  our 
simpler  illustration,  the  completed  idea  is  of  a  complex 
object  of  thought  constructed  out  of  three  constituents 
{S,  R,  T)  bound  together  in  a  certain  form  of  unity. 


*   Ji    r?*-w*.a^>a4  Hjort^  4  ^^a^jhfi^^  ^  ^..^M^  rxA^J^ 


222 


LAWS  OF  THOUGHT 


223 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


§  I.     It  has  been  customary  to  apply  the  phrase 
Laws  of  Thought  to  three  specific  formulae  ;  but  the 
application  of  the  phrase  should  be  extended  to  cover 
all  first  principles  of  Logic.    By  first  principles  we  mean 
certain  propositions  whose  truth  is  guaranteed  by  pure 
reason.    It  is  often  too  hastily  said  that  logic  as  such  is 
not  concerned  with  truth  but  only  with  consistency ;  as 
if  a  conclusion  were  guaranteed  by  formal  logic  merely 
because  it  is  consistent  with  any  arbitrarily  assumed 
premisses.    But  this  entirely  misrepresents  the  function 
of  formal  logic,  which  is  not  permissive,  but  rather 
prohibitive.     It  guarantees  the  truth — not  of  any  pro- 
position that  is  consistent  with  the  premisses — but  only 
of  the  proposition  whose  contradictory  is  inconsistent 
with  the  premisses.    And  even  this  statement  goes  too 
far ;  for  logic  does  not  allow  any  arbitrarily  chosen  pre- 
misses to  be  taken  as  true ;  and  thus  the  only  conclusions 
that  it  can  be  said  in  any  sense  to  guarantee  are  those 
which  have  been  correctly  inferred  from  premisses  that 
are  themselves  true.    When  consistency  is  placed  in  a 
kind  of  antithesis  to  truth,  it  seems  often  to  be  assumed 
that  logic  is  indifferent  to  truth.    That  the  reverse  is 
the  case  is  shown  by  the  consideration  that  to  say  that 
a  conclusion  is  validly  drawn  from  given  premisses  is 
tantamount  to  asserting  the  truth  of  a  certain  composite 
proposition,  viz.  that  the  premisses  imply  the  conclusion. 


In  enunciating  and  formulating  the  fundamental 
principles  of  Logic,  we  shall  not  enter  into  the  question 
whether  they  are  all  independent  of  one  another,  nor 
into  the  problem  as  to  how  a  selection  containing  the 
smallest  possible  number  could  be  made  amongst  them 
from  which  the  remainder  could  be  formally  derived. 
This  problem  is  perhaps  of  purely  technical  interest,  and 
the  attempt  at  its  solution  presents  a  fundamental,  if  not 
insuperable,  difficulty:  namely,  that  the  procedure  of 
deriving  new  formulae  from  those  which  have  been  put 
forward  as  to  be  accepted  without  demonstration,  is 
governed  implicitly  by  just  those  fundamental  logical 
principles  which  it  is  our  aim  to  formulate  explicitly. 
We  can,  therefore,  have  no  assurance  that,  in  explicitly 
deriving  formulae  from  an  enumerated  set  of  first 
principles,  we  are  not  surreptitiously  using  the  very 
same  formulae  that  we  profess  to  derive.  If  this  objec- 
tion cannot  be  removed,  then  the  supposition  that  the 
whole  logical  system  is  based  on  a  few  enumerable  first 
principles  falls  to  the  ground. 

§  2.  The  charge  has  been  brought  against  all  the 
fundamental  principles  of  Formal  Logic  that  they  are 
trivial ;  or  otherwise  that  they  are  nothing  but  truisms. 
Now  a  truism  may  be  defined  as  a  proposition  which  is 
(i)  true,  and  (2)  accepted  by  everybody  on  mere  inspec- 
tionas true;  and thesearejustthe  characteristics  required 
of  a  fundamental  principle  of  logic.  Hence  to  charge  the 
fundamental  formulae  with  being  mere  truisms  is  not 
to  condemn  them,  but  to  admit  that  they  are  fitted  to 
fulfil  the  function  for  which  they  are  intended.  This 
function  is  to  enable  us  to  demonstrate  further  formulae, 
some  of  which,  though  true,  are  not  accepted  by  every- 


«/.  $r 


224 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


225 


body  on  mere  inspection  as  true.  It  is  an  actual  fact 
that  by  means  of  truisms  and  truisms  alone  we  can 
demonstrate  truths  which  are  not  truisms.  The  above 
and  similar  criticisms  directed  against  the  fundamental 
formulae  of  Logic  will  be  best  met  by  directly  examining 
this  or  that  formula  so  as  to  bring  out  its  precise  signifi- 
cance in  view  of  the  different  points  of  view  from  which 
it  has  been  criticised  ;  and  we  shall  adopt  this  plan  as 
occasion  offers. 

§  3.  Before  enunciating  the  fundamental  principles 
in  detail,  we  will  enquire  into  what  is  implied  in  speaking 
of  them  as  'Laws.'  The  word  law  is  closely  connected 
with  the  notion  of  an  imperative ;  and  many  logicians 
of  the  present  day  hold  that  the  so-called  laws  of  thought 
are  no  more  imperatives  than  are  the  axioms  of  arith- 
metic or  geometry.  With  this  view  I  agree,  inasmuch 
as  the  axioms  of  mathematics  can  themselves  be  re- 
garded as  having  an  imperative  aspect ;  but  this  is 
because  all  truth  may  be  so  regarded.  The  idea  of 
truth  and  falsity,  in  my  view,  carries  with  it  the  notion 
of  an  ijnperative,  namely  of  acceptance  and  rejec- 
tion— a  corollary  from  the  theory  which  insists  on  the 
reference  of  judgment  and  assertion  to  the  thinker. 
For  it  is  only  so  far  as  assertion  is  recognised  to  be  a 
mental  act,  that  the  notion  of  an  imperative  becomes 
relevant.  An  imperative  of  reason  implies  a  restraint 
upon  the  voluntary  act  of  assertion — a  restraint  which 
does  not,  however,  infringe  the  freedom  that  charac- 
terises every  volition,  since  the  obligation  to  think  in 
accordance  with  truth  is  self-imposed.  Any  study  of 
which  imperatives  constitute  the  subject-matter  has  been 
called  a  normative  science,  and  normative  sciences  have 


been  contrasted  with  positive  sciences.     But  from  a 
certain  point  of  view  every  science  may  be  said  to 
exercise  an  imperative  function  in  so  far  as  any  mistake 
or  confusion  in  the  judgments  of  the  ordinary  man  is 
corrected  or  criticised  by  the  scientist  as  such.     Every, 
science  therefore  can  without  any  confusion  of  thought  j 
be  regarded  as  normative  ;  which  is  only  another  way  i 
of  saying  that  the  notions  of  truth  and  falsity  as  pre- 
dicate of  propositions  carry  with  them  the  notions  'to 
Ibe  accepted  '  or  *  to  be  rejected '  understood  as  impera- 
tives.   But  an  explanation  can  be  given  for  the  restricted 
ise  of  the  term  normative  to  logic,  aesthetics  and  ethics : 
Az.,  that,  while  each  deals  with  a  certain  kind  of  mental ! 
fact,  it  does  not  deal  with  it  merely ^5  fact.     Every! 
jcience  which  deals  with  man,  either  in  his  individual 
)r  social  capacity,  takes  as  its  topic  the  description  of 
lental  facts — including  an  analysis  of  how  men  think, 
feel  and  act ;  but  such  a  descriptive  study  of  our  thoughts, 
feelings  and  actions  (including  their  causal  relations) 
treated  generally,  historically  or  speculatively,  is  to  be 
listinguished  from  the  study  of  precisely  the  same  facts 
|n  relation  to  certain  norms  or  standards,  and  from  the 
:ritical  examination  of  these  norms  or  standards  them- 
>elves.    The  division  of  sciences  in  general  into  norma- 
ive  and  positive  is,  therefore,  unsound,  inasmuch  as  all 
Sciences  may  be  regarded  as  normative  in  the  sense 
Ihat  they  are  potentially  corrective  of  mistaken,  false  or 
)bscure  views.    This  division  (into  normative  and  posi- 
ive)  is  therefore  properly  restricted  to  sciences  dealing 
^ith  psychological  material ;  thus  the  positive  or  descrip- 
ive  treatment  of  mind — in  its  thinking,  feeling  or  acting 
Lspect — is  (like  all  sciences)  normative  in  the  sense  of 


a 


J.  L. 


15 


226 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


227 


A 


I. 


II. 


in. 


N, 


being  potentially  corrective  of  false  judgments  on  the 
topics  directly  dealt  with ;  while  the  treatment  in  Logic, 
Aesthetics  and  Ethics  of  these  same  processes  is  norma- 
tive in  the  more  special  sense  that  these  sciences  examine 
and  criticise  the  norms  of  thought,  feeling  or  action 
themselves.  Within  the  range  for  which  the  antithesis 
between  normative  and  positive  holds,  the  distinction 
between  a  descriptive  or  causal  account  of  psychological 
or  sociological  matters,  and  an  examination  of  standards 
or  norms,  is  now-a-days  of  the  first  importance,  inasmuch 
as  the  substitution  of  causal  description  in  the  place  of 
evaluation  of  standard  has  been  woefully  common  in 
works  which  profess  to  found  Ethics  upon  psychology 

or  sociology. 

§  4.  To  return  to  the  consideration  of  the  principles 
which  exercise  an  imperative  function.  The  funda-| 
mental  formulae  for  conjunctive  and  composite  pro- 
positions have  been  given  in  the  chapter  on  compound 
propositions;  these  must  be  included  in  the  general 
consideration  of  the  Laws  of  Thought.  Certain  of  these 
laws,  and  in  particular  the  Reiterative,  Commutative  and 
Associative  laws  of  Conjunction  are — not  only  the 
materials  which  explicitly  compose  the  logical  system— i 
but  are  also  implicitly  used  in  the  process  of  building  up 
the  system.  Thus,  for  example,  we  not  only  explicitly 
formulate  the  Reiterative  Law,  but  in  making  repeated 
use  of  this  or  of  any  other  law,  we  are  implicitly  using 
the  Reiterative  Principle  itself.  This  will  be  seen  to 
hold  in  the  same  way  of  the  Commutative  and  Associa- 
tive principles  of  Conjunction.  Finally,  inasmuch  as  thel 
system  is  developed  by  means  of  inference,  the  essen- 
tial  principles  of  implication   are  not   only  explicitly]. 


formulated  in  the  formulae  for  composite  propositions, 
bU(t  also  implicitly  used  in  constructing  the  logical 
system  itself. 

§  5.  The  next  set  of  laws  to  be  considered  will  be 
those  which  express  the  nature  of  Identity ^  since  this  is 
a  formal  conception  which  applies  with  absolute  uni- 
versality to  all  possible  objects  of  thought  whatever  the 
category  to  which  they  may  belong.  Identity  is  a  re- 
lation, and  as  such  has  certain  properties  which  are 
exhibited  in  what  we  shall  call  the  Laws  of  Identity,  3^ 
Relations  in  general  may  be  classified  according  to  the 
formal  properties  they  possess,  irrespectively  of  the 
terms  related.  It  will  be  necessary  here  to  introduce 
and  define  three  of  these  properties,  viz.,  transitiveness, 
symmetry  and  reflexiveness.  Using  the  symbols  x,  y,  z, 
to  stand  for  the  terms  of  any  relation,  and  the  symbols 
r  and  r  for  any  relation  and  its  converse,  then, 

( 1 )  the  relation  r  is  called  t£ansiti ye  :  when  'x  is  r 
toy'  and  ^yisrtoz'  together  implies  *;»;  is  f  to  ^ '  for 
all  cases  of  .^ir,  y^  z\  for  example,  ancestor,  greater  than, 
causing,  implying', 

(2)  the  relation  r  is  called  sj^mmetrical :  when  ^ x  is 
f  to^'  implies  ' y  is  r  to  x;  in  other  words,  when  'x  is 
r  toy'  implies  'x  is  r  toy,'  for  all  cases  of  x  and^;  for 
example,  cousin,  incompatidle  with,  other  than ; 

(3)  the  relation  r  is  called  reflexive:  when  'x  is  f  to 

x'  for  all  cases  of  ^;  for  example,  compatriot  of,  simul- 
taneous with,  homogeneous  with. 

Now  the  three  Laws  of  Identity  are  most  simply 
expressible  by  the  statement  that  identity  is  (i)  transi- 
tive, (2)  symmetrical,  (3)  reflexive ;  or  otherwise,  for 

15-2 


228 


CHAPTER  XIV 


I 


LAWS  OF  THOUGHT 


every  object  of  thought  (represented  by  the  symbols 
X,  y,  2). 

(i)    Transitive  Law.  \{ x  is  identical  with  j/,  andjj/ 
is  identical  with  z ;  then  x  is  identical  with  z. 

(2)  Symmetrical  Law :  If  :r  is  identical  with  y,  then 
y  is  identical  with  x, 

(3)  Reflexive  Law:  x  is  identical  with  x. 

It  will  be  observed  that  there  are  a  host  of  other  relations 
which  have  these  same  three  properties ;  e.g.  contem- 
poraneous, homogeneous,  compatriot,  numerically  equal, 
equal  in  magnitude,  etc.,  but  analysis  of  every  such  rela- 
tion shows  it  to  contain  a  reference  to  some  identical 
element,  upon  which  these  formal  properties  depend. 

§  6.  The  phrase  *  Law  of  Identity'  has  been  tra- 
ditionally used  for  one  of  the  three  fundamental  logical 
principles,  known  as  the  Laws  of  Identity,  of  Non- 
Contradiction,  and  of  Excluded  Middle,  to  which  the 
term  '  Laws  of  Thought '  has  been  usually  restricted ; 
but,  since  these  three  laws  relate  exclusively  to  pro- 
positions, whereas  the  conception  of  identity  applies  to 
all  objects  of  thought,  I  propose  to  substitute  for  the 
traditional  terminology,  the  Principles  *of  Implication,' 
*of  Disjunction'  and  'of  Alternation'  respectively  ;  and 
to  insert  a  fourth,  to  be  called  the  *  Principle  of  Counter- 
implication.'  The  four  together  will  be  entitled  'the 
i  Principles  of  Propositional  Determination.'  The  four 
laws  are  thus  brought  into  line  with  the  four  forms  of 
composite  proposition  discussed  in  a  preceding  chapter. 
The  composite  propositions  expressed  in  their  general 
form,  i.e.  in  terms  of  two  independent  components  p,  q, 
5-  are  of  course  not  guaranteed  as  true  by  pure  logic ;  in 


229 


other  words,  they  require  material  or  experiential  certifi- 
cation as  opposed  to  merely  formal  or  rational  certifica- 
tion. T\i^  principles,  on  the  other  hand,  are  those  cases  f^ 
of  the  composite  propositions,  expressed  in  their  quite  "" 
general  form,  the  truth  of  which  is  guaranteed  by  pure 
logic.  For  the  purposes  of  formulating  the  principles 
on  the  lines  of  the  four  composite  functions,  we  may 
slightly  modify  the  expression  of  these  latter  as  follows : 

(i)  Implicative  Function  :  \i  P  \s  true,  then  Q  is  true. 

(2)  Counterimplicative  Function :  If  /*  is  false,  then  Q  is  false. 

(3)  Disjunctive  Function  :  Not  both  P  true  and  Q  true. 

(4)  Alternative  Function :  Either  P  true  or  Q  true. 

T\ie,  principles  are  obtained  by  substituting  P  for  Q  in 
the  implicative  and  counterimplicative  functions,  and 
/^-false  for  ^-true  in  the  disjunctive  and  alternative 
functions.     Thus: 

Principles  of  Propositional  Determination 
{P  being  any  proposition) 

(i)  Implicative',  It  must  be  that  if  P  is  true,  then  P  is  tru^,         (  Sm^J^^ 

(2)  Counterimplicative-.  \X,  must  be  that  '\i P  is  false,  then  P  is  false, 

(3)  Disjunctive :  P  cannot  be  both  true  and  false,  (i»**^-  t^wCUJUcic^ 

(4)  Alternative-.  P  must  be  either  true  or  false,  igmd^U^  UiU^J 

where  the  words  *  must  be '  and  *  cannot  be '  serve  to 
indicate  that  the  principles  are  formally  or  rationally 
certified. 

This  formulation  uses  a  single  proposition  P  to- 
gether with  the  two  adjectives  true  and  false,  in 
preference  to  the  more  usual  mode  of  expression  which 
employs  two  propositions,  P  and  not-/*,  and  a  single 
adjective  *true';  as  in  the  following: 


230 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


231 


I 


1. 


3. 


(i)  U  P  is  true,  then  P  is  true. 
(2)  If  not-/*  is  true,  then  not-P  is  true. 
{3)  P  and  not-P  cannot  both  be  true. 
(4)  Either  P  or  not-P  must  be  true. 

There  are  several  reasons  for  adopting  the  former  of 
these  two  modes  of  formulation  in  preference  to  the 
latter.  In  the  first  place  it  uses  the  comparatively  simple 
notion  of  P  being  false  instead  of  the  rather  awkward 
notion  of  not-P  being  true.  Secondly  it  enables  us  to 
define  'contradiction'  by  means  of  the  principles,  which 
would  be  impossible  without  a  circle  if  we  introduced 
the  contradictories  P  and  not-P  into  the  formulation.  In 
the  third  place,  the  introduction  of  the  phrases  P-trut 
and  P-false  is  in  accordance  with  the  fact  that  the 
adjectives  true  and  false  are  the  first  characteristics  by 
which  the  nature  of  the  proposition  as  such  is  to  be 
understood.  A  closer  analysis  of  this  formulation  of 
the  alternative  and  disjunctive  principles  will  throw 
further  light  on  the  nature  of  the  antithesis  between 
the  adjectives  true  and  false.  We  have  emphasised  the 
point  that  these  adjectives  are  predicable  only  of  pro- 
positions; in  other  words  'anything  that  is  true  or  false 
is  a  proposition ' ;  the  principle  of  alternation  adds  to 
this  statement  its  complementary,  viz.,  *  anything  that 
is  a  proposition  is  true  or  false.'  It  is  clear,  of  course, 
that  these  two  statements  are  not  synonymous.  Again, 
the  principle  of  disjunction  states  that  the  adjectives 
true  and  false  are  incompatible ;  and  this  again  goes  be- 
yond what  is  explicitly  involved  in  the  statement  that 
they  are  predicable  exclusively  of  propositions. 

The  most  obvious  immediate  application  of  these 


principles  is  obtained  by  taking  P  to  stand  for  a  definite 
singular  proposition :  '  ^  is  /,'  where  *  s '  stands  for  a 
uniquely  determined  or  singular  subject,  and  '/'  for 
any  adjective.  Then  /'-false  becomes  '  s  is  not-/.'  In 
this  application,  the  four  principles  may  be  called  the 
Principles  of  Adjectival  Determination,  and  assume  the 
following  form : 

Principle  of  Implication :  If  j  is  /,  then  s  is  p. 

Principle  of  Counterimplication :  If  j  is  not-/,  then  s  is  not-/. 
Principle  of  Disjunction :  s  cannot  be  both  /  and  not-/. 

Principle  of  Alternation :  ^  must  be  either/  or  not-/. 

In  this  application,  the  principles  are  expressed  in  terms 
of  any  adjectives/  and  not-/  predicated  of  any  subject 
s ;  instead  of  being  expressed  in  terms  of  the  adjectives 
true  and  false  predicated  of  any  proposition  P,     In 
ordinary  logical  text-books  the  'Laws  of  Thought'  are 
almost  always  expressed  in  this  specialised  form  ;  but, 
by  this   mode    of  enunciation,   the  generality  which 
characterises  the  formulation  in  terms  of  propositipnj 
is  lost;  for  when  'adjectives  predicated  of  any  subject' 
is  substituted  for  'propositions,'  we  have  only  a  special 
case  from  which  the  general  ccfuld  not  have  been  de- 
rived.    It  is  convenient  for  many  purposes  to  use  the 
term   *  predication '  to  stand  for  'adjective'  or  'pro- 
position ' ;  thus  we  may  include  both  the  general  and  ^.^ 
the  special  formulae  of  determination  in  the  abbreviated 
forms  '  If/  then/';  '  If  not-/  then  not-/';  '  Not-both/ 
and  not-/ ' ;  'Either /  or  not-/' ;  where/  stands  for  any 
predication.    The  two  sets  of  formulae  might  again  be  ^;^ 
expressed — without  any  modification  of  meaning — in 
the  form  of  uniyersals,  since  P  stands  for  any  proposi- 
tion,  and  .y.for  any  subject;  thus: 


232 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


233 


Generalised  Form  for  Propositional 
Determination 


Generalised  Form  for  Adjectival 
Determination 


If  any  proposition  is  true,  it  is  true 
If  any  proposition  is  false,  it  is  false 
No  proposition  can  be  both  true  and  false 
Any  proposition  must  be  either  true  or  false 


If  anything  is/,  it  is  p 
If  anything  is  not-/,  it  is  not-/ 
Nothing  can  be  both/  and  not-/ 
Anything  must  be  either/  or  not-;i 


A    comparison    between    these    two    generalised 
jformulations  of  the  principles  will  bring  out  the  impor- 
tant distinction  between  false  and  not-true  and  again 
between  true  and  not-false.    According  to  the  principles 
of  adjectival  determination  '  hnythin^  must  be  either 
true  or  not-true ' ;  whereas  of  propositions  we  can  say 
'  f^'^y  proposition  must  be  either  true  or  false!    Now, 
since  it  is  only  propositions  of  which  truth  is  properly 
predicable,  therefore  of  anything  that  is  not  a  proposi- 
y    tion  the  adjective  true  must  be  denied;  thus  we  must 
say  '  The  table  is  not  true '  on  the  elementary  ground 
that  'the  table'  is  not  a  proposition;  but  we  cannot  say 
that  'The  table  is  false,'  because  it  is  only  propositions 
which  can  be  said  to  be  false.     Thus  the  principles  of 
propositional  determination  force  upon  us  the  notable 
consideration  that  the  word  false  does  not  really  mean 
the  same  as  not-true.    To  have  expressed  the  principle 
of  alternation  in  the  form  '  Anything  must  be  either 
true  or  false'  without  the  necessary  restriction  to  a  pro- 
position would  have  been  actually  wrong.    On  the  other 
hand,  the  form  '  Any  proposition  must  be  either  true 
or  not-true'  is  not  sufficiently  determinate,  for  this  alter- 
native would  hold  of  any  subject  whatever  and  fails  to 
express  the  alternative  peculiar  to  the  proposition  itself. 


On  the  same  ground,  the  disjunctive  principle  is  not 
properly  expressed  in  the  form  '  No  proposition  can  be 
both  true  and  not-true.*  This  affords  another,  and,  in 
my  view,  the  most  important  justification  for  formulating 
the  principles  in  terms  of  the  adjectives  true  and  false 
instead  of  in  terms  of  the  propositions  P  and  not-/*. 
In  illustration  of  the  above  discussion  we  may  point  to 
the  analogy  between  the  four  adjectives  true,  false,  not- 
true,  not-false  and  the  four  adjectives  male,  female,  not- 
male,  not- female.  The  antithesis  between  male  and 
not-male  or  again  between  female  and  not-female  is 
applicable  to  any  subject  whatever,  but  that  between 
male  and  female  is  applicable  exclusively  to  organisms. 
Analogously  the  antithesis  between  true  and  not-true 
or  aga^n  between  false  and  not-false  is  applicable  to  any 
subject-term  whatever,  but  that  between  true  and  false 
is  applicable  exclusively  to  propositions. 

§  7.  Now  the  so-called  Law  of  Identity — which  I 
have  expressed  in  the  form  *  If  P  is  true  then  P  is  true' 
where  P  stands  illustratively  for  any  proposition — has 
been  the  favourite  object  of  attack  by  critics  of  the  prin- 
ciples of  formal  logic,  on  the  score  of  its  insignificance 
or  triviality.  The  reason  why  the  formula  appears  to 
have  little  or  no  significance  is  that  its  implicans  is 
literally  identical  with  its  implicate.  It  will  be  found, 
however,  that  the  necessary  condition  for  the  explicit 
use  of  the  relation  of  identity  is  that  the  identified  ele- 
ment should  have  entered  into  different  contexts.  It 
must  be  noted  that  this  necessary  reference  to  difference 
of  context  does  not  render  the  relation  of  identity  other 
than  absolute,  i.e.  it  in  no  way  implies  that  in  its  two 
occurrences  the  identified  element  is  partly  identical 


,#  s 


234 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


235 


I  and  partly  dififerent.  The  application  of  this  general 
I  condition  to  the  Principle  of  Implication  requires  us  to 
contemplate  the  proposition  ' P  is  true'  as  one  that  may 
have  been  asserted  in  different  connections  or  on 
different  occasions  or  by  different  persons.  Then,  since 
the  formula  *If  /^  is  true  then  P  is  true'  is  to  be  under- 
stood as  logically  general,  its  full  import  can  be  expressed 
in  the  form:  *  If  the  asserting  of  P  in  any  one  context 
/  is  true,  then  the  asserting  of  P  in  any  context  whatever 
is  true.'  If  this  analysis  be  accepted,  it  will  be  found 
that  the  principle  could  not  have  been  enunciated  except 
for  the  possibility  of  identifying  an  assertum  or  pro- 
position as  distinct  from  the  various  attitudes  (belief, 
interrogation,  doubt,  denial)  which  might  have  been 
adopted  towards  it  on  different  occasions  by  the  same 
or  different  persons.  One  important  element  of  meaning, 
therefore,  implicit  in  the  formula  is  that  it  tacitly  implies 
the  identifiability  of  a  proposition  as  such. 

Turning  now  to  the  adjective  *  true '  as  it  occurs  in 
our  analysis  of  the  formula,  let  us  contrast  it  with  cer- 
tain adjectives  that  are  predicable  of  things  in  general. 
The  principle — that  what  can  be  asserted  in  one  con- 
text as  true  must  be  asserted  in  any  other  context  as 
true — is  more  familiarly  particularised  in  the  form  'any 
^  proposition  that  is  once  true  is  always  true ' ;  that  is  to 
■  say  that  *  true '  as  predicable  of  any  proposition  is  un- 
I  alterable ;   whereas  there  are  certain   adjectives   and 
relations  predicable  of  things  in  general  which  may 
characterise  them  only  temporarily.     Contrasting,  for 
instance,  the  Principle  of  Propositional  Determination 
*  If  any  proposition  is  true  it  is  true'  with  the  Principle 
of  Adjectival  Determination  *  If  anything  is  p  it  is  /,' 


we  find  that  in  the  former  the  copula  'is'  is  to  be  inter- 
preted without  reference  to  the  present  or  any  other 
assigned  time ;  whereas  in  the  latter  the  adjective/  may 
be  alterable,  so  that  the  copula  4s'  must  here  be  under- 
stood as  referring  to  definitely  assigned  time.  In  the 
case  of  anything  that  is  at  an  assigned  moment  of  time 
/,  the  principles  of  logic  do  not  entitle  us  to  assert  that 
it  will  be  or  has  always  been  p.  Taking  as  examples 
'The  water  has  a  temperature  of  30°  C  or  ^Mr  B.  is  at 
home,'  we  must  say  on  the  one  hand  that  if  these  pro- 
positions are  true  at  any  time,  they  are  true  at  all  times,  a. 
But  we  must  not  say  that  if  the  predicate  '  having  a 
temperature  of  30°  C  or  the  relation  'being  at  home'  ^^ 
is  true  of  a  given  subject  at  one  time,  it  will  be  true  at 
all  times.  This  obvious  comment  would  not  have  been 
required  if  language  had  distinguished  in  the  mode  of 
the  verb  'to  be '  between  a  tuneless  predication  and  a 
tense  (present,  past  or  future).  Certain  logicians  have, 
however,  deliberately  denied  the  dictum  that  what  is 
once  true  is  always  true,  and  their  denial  appears  to  be 
due  to  a  confusion  between  the  time  at  whkh  an  asser- 
tion is  made,  and  the  time  to  which  an  assertion  refers; 
or  as  Mr  Bosanquet  has  neatly  put  it— 'between  the 
time.^/predication  and  the  tune  in  piedication.'  Others, 
i.e  the  Pragmatists,  have  made  the  denial  of  this  dictum 
a  fundamental  factor  in  their  philosophy,  inasmuch  as 
they  have  taken  the  term  'true'  to  be  virtually  equiva- 
lent to  'accepted,'  whereas  everybody  else  would  agree 
that  the  term  is  equivalent  rather  to  the  phrase  '  toje 
accepted.'  Again,  the  dictum  would  not  have  been  con- 
fidently admitted  in  the  days  before  the  principles  of 
Logic  had  been   formulated   by  Aristotle,   when  the 


^ 


J^i^^ll'^(W^!l^Wli^F^^^P| 


.'/I 


236 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


237 


antithesis  between  the  immutability  of  truth  and  the 
mutability  of  things  appears  to  have  presented  an  in- 
surmountable problem.  Since,  then,  it  has  been  disputed 
from  three  different  points  of  view  that  the  truth  or 
falsity  of  a  proposition  is  independent  of  the  time  of 
assertion,  the  first  Law  of  Thought — my  interpretation 
of  which  brings  out  clearly  this  quality  of  truth — is 
effectively  freed  from  the  charge  of  triviality. 

But  not  only  must  we  interpret  the  principles  as 
implying  the  unalterability  of  truth,  for  further,  according 
to  the  principle  of  disjunction,  a  proposition  cannot  be 
both  true  and  false;  and  this  is  to  be  interpreted  to  im- 
ply that,  if  a  proposition  is  true  in  any  one  sense,  there 
can  be  no  sense  of  the  word  true  in  which  it  could  be 
I  false,  or  other  than  true.  On  this  interpretation  the 
principle  would  be  opposed  by  those  philosophers  who 
employ  the  words  rejative  and  absolute,  or  similar  terms, 
to  distinguish  two  kinds  of  truth.  In  consistency  with 
this  philosophical  position,  the  term  *true'  must  be  said 
to  have  two  meanings,  so  that  one  and  the  same  pro- 
position might  be  true  in  one  meaning  of  the  term  *true' 
and  false  in  another.  It  would  seem  that  it  is  only  on 
this  theory  that  philosophers  could  maintain  that  a  cer- 
tain proposition  such  as  '  Matter  exists'  is  true  in  or  for 
science,  and  at  the  same  time  false  from  the  point  of 
view  of  philosophy.  According  to  the  view,  however, 
of  those  who  maintain  rigidly  the  validity  of  the 
Principles  of  Determination,  it  cannot  be  said  that  the 
same  proposition  is  true  in  one  sense  and  false  in  another 
sense,  although  it  may  be  said,  of  course,  that  one  sense 
given  to  a  certain  collocation  of  words  would  yield  a 
true  proposition,  while  another  sense  given  to  the  same 


collocation  of  words  would  yield  a  false  proposition. 
We  must  deny  for  instance  that  'Matter  exists'  can  be 
true  in  one  sense  and  false  in  another  sense,  though  we 
do  not  for  a  moment  dispute  that  *  Matter  exists  *  in 
one  sense  may  be  true  while  'Matter  exists'  in  another 
sense  may  be  false.  It  is  noteworthy  that  the  confusion 
here  is  exactly  parallel  to  that  between  the  time  of  pre- 
dication and  the  time  in  predication.  Thus  the  assertion 
'Mr  Brown  is  at  home'  cannot  b£  true  at  one  time  and 
false  at  another;  though  that  'Mr  Brown  is  at  home  at 
one  time'  may  be  true  and  that  *Mr  Brown  is  at  home 
at  some  other  time '  may  be  false. 

§  8.  We  propose  now  to  consider  the  Principles  of 
Adjectival  Determination  with  a  view  to  giving  added 
significa<nce  to  the  predicational  factor  by  bringing  out 
the  relation  of  an  adjective  to  its  determinable.  For  this 
purpose  the  principles  will  be  reformulated  as  follows : 

(i)  Principle  of  Implication;  If  ^  is/,  where/ is  a 
comparatively  determinate  adjective,  then  there  must 
be  some  determinable,  say  P,  to  which  /  belongs, 
such  that  s  \^  P, 

(2)  Principle o/Counterimplication:  Ifi" is/*,  where 
/  is  a  determinable,  then  s  must  be  /,  where  /  is  an 
absolute  determinate  under  P, 

(3)  Principle  of  Disjunction',  s  cannot  be  both  / 
and  /',  where  /  and  /'  are  any  two  different  absolute 
determinates  under  P, 

(4)  Principle  of  Alternation-.  ^  must  be  either  not-/*; 
or/  or/' or/''...  continuing  the  alternants  throughout 
the  whole  range  of  variation  of  which  /  is  susceptible — 
/,/',/"...  being  comparatively  determinate  adjectives 
under  P. 


238 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


239 


For  convenience  of  reference,  these  formulae  may 
be  elliptically  restated  as  follows: 

(i)  Us  is^,  then  s  is  P. 

(2)  If  s  is  /*,  then  s  is  p. 

(3)  s  cannot  be  both/  and/',  nor/'  and/",  nor/  and/".... 

(4)  s  must  be  either  not-/*  or/  or/'  or/"  or/'".... 

Contrasting  this  reformulation  with  the  original 
formulation  of  the  principles  of  adjectival  determina- 
tion, it  will  be  obser\i^ed  that,  while  the  predications  of 
s  are  more  precise,  they  are  not  so  palpably  obvious. 

/.  The  force  of  the  first  principle  is  that,  if  a  subject  is  of 
such  a  kind  that  a  certain  determinate  adjective  can  be 
predicated  of  it,  then  this  presupposes  that  the  subject 
belongs  to  a  certain  category  such  that  it  may  be  com- 
pared in  character  with  other  subjects  belonging  to  the 
same  category,  the  ground  of  comparison  being  equiva- 

j£^  lent  to  the  determinable.  The  second  principle  states 
that  any  subject  whose  character  is  so  far  known  that 
a  certain  determinable  adjective  can  be  predicated  of 
it,  must  in  fact  be  characterised  by  some  absolutely 
determinate  value  of  that  determinable,  and  that,  al- 
though, in  many  cases,  such  a  precise  determination  of 
character  is  impossible,  yet  the  postulate  that  in  fact 
the  subject  has  some  determinate  character  is  one  that 

i]|. reason  seems  to  demand.  The  third  principle  as  re- 
formulated gains  in  significance,  as  compared  with  the 
mere  disjunction  of  p  with  the  indeterminate  not-/, 
since  now  it  precludes  the  possibility  of  conjoining  an 
indefinite  number  of  pairs  of  predicates,  which  are 
here  exhibited  as  determinate  and  positive.  In  fact,  in 
the  principle  of  disjunction  in  its  original  form  (accord- 
ing to  which/  cannot  be  joined  with  not-/)  not-/  should 


signify — not  merely  some  or  any  adjective  other  than 
p — but  some  adjective  that  is  necessarily  incompatible 
with  /,  and  the  only  such  adjectives  are  those  other 
than  /  which  belong  to  the  same  determinable. 

The  significance  of  the  Principle  of  Alternation  in  ^ 
its  new  form  requires  special  discussion.  It  is  developed 
from  the  dichotomy  'Any  subject  must  be  either  not-/* 
or  P'  where  P  stands  for  any  determinable.  This 
again  assumes  that  'Some  subjects  are  not  /*,'  i.e.  that 
there  are  subjects  belonging  to  such  a  category  that 
the  determinable  adjective  P  is  not  predicable  of  them. 
The  negative  here  must  be  termed  a  pure  negative,  in 
the  sense  that  not-/*  cannot  be  resolved  into  an  alter- 
nation of  positive  adjectives.  For  example,  in  the 
statement :  '  Material  bodies  are  not  conscious '  the 
negative  term  '  not  conscious '  does  not  stand  for  any 
single  positive  determinable  which  would  generate  a 
series  of  positive  determinates.  We  ought  in  fact  to 
maintain  that  'not-conscious*  is  not  properly  speaking  an 
adjective  at  all ;  for  in  accordance  with  the  reformulated 
Principle  of  Adjectival  Implication,  every  adjective 
that  can  be  predicated  of  a  subject  must  be  a  more  or 
less  determinate  value  of  some  determinable  \  Elimin- 
ating, then,  the  negative  not-/*  from  the  predicate, 
the  reformulated  Principle  of  Adjectival  alternation 
may  now  be  expressed  in  the  form  :  *  Any  subject  that 
is  P  must  be  either  /  or/'  or/"  or... '  where  the  alter- 

^  A  negative  predication  of  this  type  has  sometimes  been  called 
Privative)  but  unfortunately  the  term  privative  has  also  been  used  in 
an  opposite  sense,  namely,  for  a  predication  applied  to  a  subject 
belonging  to  a  category  for  which  the  positive  adjective  is  normally  ^ 
applicable ;  as  when  we  predicate  of  a  person  that  he  is  blind  or  that 
he  is  (temporarily)  unconscious. 


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241 


native  predication  /  or  /'  or  /''  or  . . .  is  restricted  to 
subjects  of  which  the  determinable  P  is  predicable.    If 
we  compare  this  form  of  the  Principle  of  Alternation 
with  the  Principle  of  Counterimplication,  viz.,   If  s  is 
characterised  by  P,  it  must  be  characterised  by  one  or 
other  determinate  value  of  P,  there  would  appear  to 
be  no  obvious  difference  between  them.    The  Principle 
of  Alternation,  however,  supplements  that  of  Counter- 
implication  by  implicitly  postulating  that  the  range  of 
possible  variation  of  the  determinable  can  be  appre- 
hended in  its  completeness.    The  question  whether  we 
can    in  this  way  apprehend    the    complete   range   of 
possible  variation  of  any  determinable  must  be  ex- 
amined in  detail.    Consider  the  determinable  '  integral 
number '  which  is  always  predicable  of  a  collection  or 
aggregate  as  such.    Of  a  'collection'   we  can   in  the 
first  place  assert  universally  that  'it  is  either  zero  or 
greater  than  zero,'  and  this  it  is  to  be  observed,  goes 
beyond    the    mere   assertion   that   it  is    'either   zero 
or  not  zero.'   Again  we  may  assert  that  any  collection 
is  'either  zero  or  one  or  more  than  one,'  where  the 
alternant  '  more  than  one '  is  not  merely  negative,  but 
positive — though    comparatively  indeterminate.     Pro- 
ceeding in  this  way,  we  may  resolve  exhaustively  the 
range  of  possible  variations  of  number  by  an    enu- 
merated and  finite  series  of  positively  indicated  alter- 
nants :  '  zero  or  one  or  two  or  three  or... or  n  or  greater 
than  n!   What  is  here  said  of  integral  number  holds  of 
quantity  in  general,  and  may  also  be  applied  to  any 
L  determinable  (continuous  or  discrete)  whose  determi- 
I  nates  have  an  order  of  betweenness  and  can  therefore 
I  be  serially  arranged.    For  example,  the  range  of  hue 


can  be  exhaustively  resolved  into  the  nine  alternants 
*  red,  or  between  red  and  yellow,  or  yellow,  or  between 
yellow  and  green,  or  green,  or  between  green  and  blue, 
or  blue,  or  between  blue  and  violet,  or  violet.' 

We  may  summarise  (with  some  additional  comments) 
what  has  been  said  with  respect  to  the  Principles  of 
Adjectival  Determination,  formulated  with  reference 
to  the  determinable,  (i)  If  ^  is  /,  then  s  is  P,  This 
postulates  that  whenever  a  comparatively  determinate 
predication  is  asserted,  then  a  determinable  to  which 
the  determinate  belongs  can  always  be  found ;  but  it 
must  be  pointed  out  that  language  does  not  always 
supply  us  with  a  name  for  the  determinable.  (2)  If  s 
is  P,  then  s  is  /.  This  postulates  that  in  actual  fact 
every  adjective  is  manifested  as  an  absolute  determin- 
ate ;  it  is  to  be  supplemented,  however,  by  the  recog- 
nition that  for  a  continuously  variable  determinable  it 
is  impossible  actually  to  characterise  a  given  subject  by 
a  precisely  determinate  adjective.  (3)  ^  cannot  be  both/ 
and  /'.  This  asserts  that  any  two  different  determin- 
ates are  incompatible;  but,  inasmuch  as  we  are  unable 
practically  to  characterise  an  object  determinately  (in 
the  case  of  a  continuously  variable  determinable),  we 
must  apply  the  formula  to  the  case  where  /  and  /' 
(though  only  comparatively  determinate)  are  figura- 
tively speaking  'outside  one  another.'  To  represent 
this  figurative  analogy,  suppose  a  point  (a,  b,  c  or  cl) 
to  represent  an  absolute  determinate,  and  the  segment 
of  a  line  (/  or  /')  to  represent  a  comparative  deter- 
minate : 


/ 


a 


J.  L. 


16 


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CHAPTER  XIV 


LAWS  OF  THOUGHT 


243 


then,  if  b  is  between  a  and  c,  and  only  then,  can  we 
assert  that  the  (comparative)  determinates  p  and  /' 
are  codisjunct  or  incompatible.  (4)  Any  s  that  is  P 
must  be  either/  or/'  or..,.  Here  the  predications 
py  P^y  P"^  etc.  need  not  be  absolute  determinates,  but 
to  render  the  principle  practically  significant  it  is 
necessary  that  we  should  be  able  to  compass  in  thought 
the  entire  stretch  or  range  of  variation  of  which  P  is 
susceptible. 

§  9.  We  will  now  pass  to  the  principles  according 
to  which  the  manifested  value  of  any  one  variable  is 
determined  by  its  connection  with  the  manifested  values 
of  other  variables.  These  principles  may  be  expressed 
in  forms  analogous  to  those  of  adjectival  determination 
and  will  be  entitled  the  Principles  of  Connectional  L^e- 
^£^^i'^oUion,  They  embody  the  purely  logical  properties 
of  the  causal  relation  ;  but  the  notion  of  cause  and 
effect — being  properly  restricted  to  phenomena  tem- 
j  porally  alterable — will  be  replaced  by  the  wider  notion 
I  of  determining  and  determined.  The  characters  which 
may  be  said  jointly  to  determine  some  other  character 
correspond  to  what  is  commonly  called  the  cause,  while 
any  character  which  is  thereby  determined  corresponds 
to  an  effect.  Analysis  of  the  general  conception  of 
causal  connection  reveals  two  complementary  aspects 
which  may  be  thus  expressed  :  {a)  wherever,  in  two 
instances,  there  is  complete  agreement  as  regards  the 
cause-factors,  there  will  be  agreement  as  regards  any 
effect-factor ;  and  {b)  wherever,  in  two  instances,  there 
is  any  (partial)  difference  as  regards  the  cause-factors, 
there  will  be  some  difference  in  one  or  other  of  the 
effect-factors.     In  formulating  the  Principles  of  Con- 


nectional  Determination,  such  symbols  as  P,  Q,  R,  T, 
will  be  introduced  to  represent  the  characters  that  are 
connectionally  determined,  along  with  A,  B,  C,  D,  to 
represent  those  which  connectionally  determine  the 
former.  Thus  the  conjunction  abed  would  correspond 
to  a  cause-complex,  and  pqrt  to  an  effect-complex. 

Principles  of  Connectional  Determination 

(i)  Principle  of  Implication,  Taking  any  deter- 
minable P,  the  determinate  value  which  it  assumes  in 
any  manifestation  is  determined  by  the  conjunction  of 
a  finite  number  of  determinables  A,  B,  C,  D  (say), 
such  that  any  manifestation  that  has  the  determinate 
character  abed  (say)  will  have  the  determinate  character 
/(say).    < 

(2)  Principle  of  Counterimplication.  Taking  any 
determinable  Ay  the  determinate  value  which  it  assumes 
in  any  manifestation  determines  (in  conjunction  with 
other  factors)  a  conjunction  of  a  finite  number  of  de- 
terminables P,  Q,  i?,  T  (say),  such  that  if,  for  instance, 
some  manifestation  having  the  determining  character 
a  has  the  determined  character  pqrt,  then  any  mani- 
festation that  has  the  (different)  character  a'  will  have 
one  or  other  of  the  different  characters  /'  or  q'  or  r^  or 

i'  (say). 

(3)  Principle  of  Disjunction,  P  being  one  of  the 
characters  determined  by  the  conjunction  of  the  deter- 
mining characters  A,  B,  C,  D,  there  can  be  no  three 
instances  characterised  respectively  by 

abcd'^py  a'bcd'^p',  a^bcd*^  p, 

(4)  Principle  of  Alternation,  On  the  same  hypo- 
thesis,  it   must  be  that  either  '  every  abed  is  p'  or 

16 — 2 


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CHAPTER  XIV 


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245 


*  every  adccl  is  /"or  *  every  adcd  is  /'' '  or,  etc.,  where 
the  range  of  alternation  covers  all  possible  determinate 
values  of  P, 
1.  The  Principle  of  Implication  postulates  that  the 
determinate  value  assumed  by  any  variable  is  depen- 
dent in  any  instance  not  upon  an  indefinite  number  of 
conditions  which  might  in  some  sense  be  exhaustive 
\  of  the  whole  state  of  the  universe,  but  upon  a  set  of 
]  conditions  that  are  capable  of  enumeration.  The  theo- 
retical and  practical  possibility  of  enumerating  the 
factors  which  together  constitute  the  determining  com- 
plex, enables  us  to  express  the  nature  of  reality  in 
universal  propositions  of  the  form  *  every  adcd  is  /.' 
If  the  character/  could  be  predicated  universally  only 
of  a  class  determined  by  an  infinite  number  of  conjoined 
characters,  reality  could  not  be  described  by  means  of 
universal  propositions  ;  or  in  other  words,  nature  would 
not  present  uniformities  which  could  be  comprehended 
•»  by  thought;  in  short  there"  would  be  nothing  that 
i  could  be  called  Laws  of  Nature.  Hence  the  signifi- 
cance of  our  first  principle  is  that  reality  presents 
uniformities  that  can  be  comprehended  in  thought,  and 
that,  whatever  variable  aspect  of  the  universe  we  may 
be  concerned  with,  a  uniformity  or  law  could  be  found 
such  that  from  it  the  value  of  the  variable  in  any 
manifestation  could  be  inferred  from  knowledge  (at 
least  theoretically  possible)  of  the  values  assumed  by 
other  variables.  The  Principle  of  Implication  repre- 
sents that  more  familiar  aspect  of  the  so-called  Law  of 
Causation  expressed  in  terms  of  agreement :  that  in 
any  two  instances  where  there  is  complete  agreement 
as  regards  the  cause  complex,  there  will  be  agreement 


as  regards  the  effect ;   or,  still  more  colloquially,  the 
same  cause  entails  the  same  effect. 

Turning  now  to  the  Principle  of  Counterimplication,   jr^ 
this  represents  the  other  and  complementary  aspect  of 
causation ;  namely  that  of  difference.    It  postulates  that 
we  can  by  enumeration  exhaust  the  characters  that  are 
determined  in  their  variation  by  any  cause  complex; 
just  as  we  assumed  that  the  cause  complex  in  the  pre- 
vious  principle  could   be    exhaustively  described.     In 
other  words  the  effect,  determined  by  any  variation  in 
the  causal  or  determining  complex,  does  not  permeate 
the  whole  universe,  but  is  restricted  to  some  assignable 
sphere.    This  important  postulate  being  presumed,  the 
principle  proceeds  to  state  that,  if  any  variable  presents 
a  different  value  in  two  instances,  indications  of  this 
difference  will  be  shown  in  one  or  other  of  the  variables 
that  are  affected  or  determined  by  the  given  variable. 
This  principle  is  therefore  complementary  to  the  pre- 1 
ceding  one;  whereas  the  Principle  of  Implication  asserts  I 
that  where  there  is  agreement  in  the  cause  there  will 
be  agreement  in  the  effect,  the  Principle  of  Counter- 
implication  asserts  that  where  there  is  difference  in  the 
cause  there  will  be  a  difference  in  the  effect.     It  may 
perhaps  even  be  said  that  in  the  popular  conception  of 
cause  this  latter  aspect— viz.  of  difference— is  more  pro- 
minent than  the  former,  viz.  agreement.    Here  we  must 
point  out  that  the  principles  are  not  parallel,  inasmuch 
as  complete  agreement  in  the  cause  is  required  to  ensure 
agreement  in  the  effect,  whereas  any  partial  difference 
in  the  cause  will  entail  some  difference  in  the  effect. 

A  word  must  be  said  about  the  strictly  formal  relations 
between  these  Principles  of  Implication  and  Counterim- 


246 


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247 


L 


plication.    By  what  is  familiarly  known  as  inference  by 
contraposition,  the  proposition  *  Every  abed  is/'  is  equi- 
valent to  the  proposition  '  Every  p'  is  either  a'  or  b'  or 
c'  or  d\'  Similarly  the  proposition  *  Every  a'  is  p'  or  q 
or  r'or  f'  is  equivalent  to  *  Every /^r/  is  a'    Applying 
this  formal  contraposition  to  the  formulae  for  cause  and 
effect,  we  see  that  the  proposition  that  *The  same  cause 
always  entails  the  same  effect'  is  logically  equivalent  to 
*Any  difference  in  the  effect  would  entail  some  differ- 
ence in  the  cause ' ;  and  again  the  proposition  that  'Any 
difference  in  the  cause  will  entail  some  difference  in  the 
effect' js  logically  equivalent  to  'The  same  effect  always 
entails  the  same  cause/    It  will  be  thus  seen  that  the 
implicative  and  counterimplicative  principles  are  not  ob- 
tainable one  from  the  other  as  equivalents  by  contra- 
position, but  are  complementary  to  one  another,  so  that 
taken  together  they  represent  the  relation  between  cause 
and  effect  as  reciprocal.    Take  the  one  aspect  of  this 
relation;  then  plurality  of  cause  holds  in  the  sense  that 
the  effect  may  be  partially  the  same  in  two  instances 
where  the  cause  is  different;    and  plurality  of  effects 
holds  in  the  same  sense,  namely,  that  the  cause  may 
be  partially  the  same  in  two  instances  where  the  effect 
is  different.   Take  the  other  aspect  of  the  relation :  thus, 
when  the  effect  is  completely  and  determinately  char- 
acterised the  character  of  the  cause  is  thereby  uniquely 
determined,  just  as  when  the  cause  is  completely  and 
determinately  characterised  the  character  of  the  effect 
is  thereby  uniquely  determined.    Thus,  whether  we  are 
considering  the  relation  of  cause  to  effect  or  of  effect  to 
cause,  the  principles  postulated  will  be  in  terms  of  com- 
plete agreement  or  of  partial  difference. 


We  pass  now  to  the  Disjunctive  Principle.    In  order 
to  expound  this  we  must  consider  three  instances  of 
A  BCD  which  agree  as  regards  the  determinate  values 
of  all  but  one,  viz.  A,  of  these  determinables.    Then, 
taking  into  consideration  the  Counterimplicative  Prin- 
ciple, a  difference  as  regards  A  in  two  instances  would 
entail  some  difference  in  one  or  other  of  the  characters 
that  are  determined  by  the  complex  ABCD.    The  prm- 
ciple  then  states  that,  if,  in  some  pair  of  instances,  a 
variation  in  the  determining  factor  A  entails  a  variation 
in  the  selected  character  P,  then  any  further  variation 
in  A  would  entail  a  further  variation  in  this  same  char- 
acter P\  whereas  if,  in  two  instances,  a  variation  in 
A  entails  no  variation  in  P,  then  any  further  variation 
in  A  would  entail  no  further  variation   in  the  same 
character  P.    It  is  essential  to  note  that  the  Disjunctive 
Principle  could  not  have  been  formulated  as  a  disjunction 
of  two  types  of  instance,  such  as  abcd~p  and  a"bcd~p. 
This  disjunction  would  be  equivalent  to  asserting  that 
a  variation  in  any  determining  factor  such  as  A  would 
entail  a  variation  in  any  or  every  determined  factor  such 
as/*;  whereas  the  Counterimplicative  Principle  has  laid 
down  only  that  a  variation  in  A  would  entail  a  variation 
in  one  or  other  of  the  determined  characters  and  not 
necessarily  in  every  one  of  them.  The  Principle  of  Dis- 
junction then  supplements  that  of  Counterimplication 
by  maintaining  that  if  some  one  variation  in  A  entails  a 
variation  in  the  selected  character  P,  then  any  variation 
in  A  would  entail  a  variation  in  the  same  character  P. 
It  might  be  supposed,  in  the  case  where  a  variation 
of  A  entails  no  variation  in  P,  that  P  is  not  causally 
connected  with  A,  and  that  therefore  A  could  be  elimi- 


lE. 


248 


CHAPTER  XIV 


LAWS  OF  THOUGHT 


249 


W 


fi 


nated.  But  the  mere  elimination  of  A  is  not  in  general 
permissible,  since  the  character  P  in  some  one  of  its 
determinate  values  requires  that  A  should  be  manifested 
in  some  or  other  of  its  determinate  values;  though,  as 
regards  the  determinate  value  of  P,  it  may  be  a  matter 
of  indifference  what  specific  value  A  has.  Since  in  this 
case  A  cannot  be  eliminated,  it  would  be  symbolically 
requisite  to  express  the  relation  of  determination  for  the 
case  under  consideration — not  in  the  form  '6cd  deter- 
mines/'— but  in  the  form  'Adcd  determines  /,'  where 
the  significance  of  the  symbol  A  is  that  any  determinate 
value  may  from  instance  to  instance  be  manifested  with- 
out affecting  the  determinate  value  /. 

Fourthly,  the  Alternative  Principle  of  Connectional 
Determination,  asserts  an  alternation  of  universal  pro- 
positions, and  of  course  goes  beyond  any  statement 
that  could  be  derived  from  the  Principle  of  Adjectival 
Alternation,  in  which  the  alternative  is  in  the  predicate. 
Thus  the  latter  states  the  universal  proposition  that 
*  Every  a^^^  is/ or/ or/' or  ...'whereas  the  principle 
under  present  consideration  states  an  alternation  be- 
tween the  universal  propositions  'Every  adcd  is  /'  or 
'Every  abcdx^p"  or.... 

These  principles  will  be  very  much  more  fully  dis- 
cussed when  we  deal  with  the  topic  of  formal  or  demon- 
strative induction ;  they  have  been  introduced  at  this 
early  stage  of  our  logical  exposition  in  order  to  indicate 
the  nature  of  the  transition  from  the  Principles  of  Pro- 
positional  Determination  which  are  purely  axiomatic,  to 
those  of  Adjectival  Determination  under  a  determinable, 
which  have  the  character  partly  of  axioms  and  partly 
of  postulates,  and  from  these  again  to  the  Principles  of 


Connectional  Determination  which  may  be  taken   as 
pure  postulates. 

§  10.    The  formulation  of  the  principles  of  connec- 
tional determination    has   an  important  bearing  upon 
the  problem  of  internal  and  external  relations.    In  con- 
troversies  on  this  topic  it  appears  to  be  agreed  that  the 
division  of  relations  into  internal  and  external  is  both 
exclusive  and  exhaustive;  and  yet  there  seems  to  be 
no  agreement  as  to  what  precisely  the  distinction  is. 
One  school  holds  that  all   relations  are  internal;  the   ^ 
other  that  all  are  external.   But  on  the  face  of  it  it  would  J.^^^^ 
appear  that  some  must  be  internal,  others  external  ;  y 
for  otherwise  it  would  seem  impossible  to  give  meaning 
to  the  distinction.   It  will  be  found,  however,  that  those 
who  deny  external  relations  doubt,  for  instance,   not    \ 
whether  spatial  and  temporal  relations  are  properly  to 
be  called  external,  but  rather  whether  space  and  time 
are  themselves  real  in  the  sense  that  the  real  can  be 
truly  characterised  by  spatial  and  temporal  relations  ; 
those  on  the  other  side  who  deny  internal  relations  ^. 
apparently  hold  that  the  independent  otherness  of  the 
terms  of  the  relation   renders  the    relation   external, 
inasmuch  as  the  specific  and  variable  relation  of  one 
term  to  another  is  not  that  which  determines  or  is 
determined  by  the  mere  existence  of  the  one  or  of  the 
other   term.    The  adherents  then  of  the  exclusively 
internal  view  of  relations  hold  that  the  relation  and  its^ 
termTare  mutually  determinative,  and  the  adherents  of 
the  exclusively  external  view,  that  the  relation  and  its 
terms  are  mutually  non-determinative  or  independent. 
Now  it  appears  to  me  that  the  root  misunderstanding 
amongst  the  two  schools  of  philosophy  on  this  point  is, 


250 


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251 


not  as  to  what  is  meant  by  an  internal  as  contrasted 
with  an  external  relation,  but  rather  what  is  the  nature 
of  the  terms  between  which  the  relation  is  supposed  to 
subsist.  The  one  school  maintains  that  the  relation 
subsists  between  the  characters  of  the  two  related 
terms  ;  the  other  that  it  subsists  between  the  terms 
themselves.  According  to  the  former  contention,  rela- 
tions are  internal  in  the  sense  that  they  depend  wholly 
upon  the  character  of  the  terms  related ;  according  to 
the  latter,  they  are  external  in  the  sense  that  they  do 
not  depend  at  all  upon  the  mere  existence  of  the  terms 
g!ua  existents.  In  this  connection  there  is  a  further 
source  of  confusion,  namely  as  to  whether  in  the 
character  of  a  term  are  to  be  included  such  relations 
as  those  of  space  and  time,  these  being  admittedly 
external,  in  contrast  to  qualities  proper  which  are 
admittedly  internal. 

At  this  point  I  will  state  my  solution  of  the  problem, 
which  will  appear  so  simple  that  it  would  seem  difficult 
to  account  for  the  origin  of  the  controversy.  I  hold, 
then,  that  relations  between  adjectives  as  such  are 
internal ;  and  those  between  existents  as  such  are 
external.  In  this  account,  adjectives  are  to  include  so- 
called  external  relations,  even  the  characterising  relation 
itself,  as  well  as  every  other  relation.  The  otherness 
which  distinguishes  the  'this'  from  the  'that'  is  the 
primary  and  literally  the  sole  external  relation,  being 
itself  direct  and  underived.  And  this  relation  is  involved 
in  every  external  relation.  In  fact,  qua  existent,  the 
'this'  and  the  'that'  have  no  specific  relation.  The 
specific  external  relations  that  hold  of  one  to  another 
existent  are  derivative  from  their  characters,  in  the 


wider  sense  of  character.    Thus,  the  relation  of  the 
'this'  to  the  'that'  obtained  from  the  fact  that  'this 
is  blue  and  that  is  green,'  is  derived  from  the  nature 
of  the  qualities  blue  and  green.    Again  the  relation  of 
proximity  or  remoteness  obtained  from  the  fact  that 
'this  is  here  and  that  is  there,'  is  derived  from  the 
positions  of  the  'this'  and  the  'that'  by  which  their 
specific  spatiality  is  characterised.  The  most  important 
application  of  the  distinction  is  to  causal  and  other 
forms  of  connectional  determination.   Here  the  primary 
relation  called  cause  is  that  between  the   character, 
dating,  and  locating,  of  two  occurrences,  from  which 
the   relation   between   the  occurrences  themselves  is 
derived,,  the    former    being    internal    and   the   latter 
external.   If  there  were  no  such  internal  causal  relation, 
nothing  could  be  stated  as  to  the  relation  of  event  to 
event,  except  that  the  one  is  invariably  accompanied 
by  the  other  in  a  certain  assignable  spatial  relation  of 
space  and   time;  and   even    this  external   relation    is 
derived  from  the  internal  relation  subsisting  between  ^ 
the  temporal  and  spatial  positions  occupied  by  the  two 
events.   If,  however,  all  spatial  properties  were  relative, 
as  is  maintained  by  Einstein  and  his  followers,  there 
would  be  no  spatial  relations  other  than  internal,   in 
fact  nothing  to  distinguish  a  space  from  that  which 
occupies  it.  The  principles  of  connectional  determination 
have  therefore  been  expressed  directly  in  terms  of  the 
characters  by  which  the  manifestations  of  reality  may 
be  described,  from  which  must  be  derived  the  external 
relations  between  such  manifestations  themselves.    It| 
will  have  been  observed  that  the  correlative  notions  of 
determination  and  dependence  enter  into  the  formulation  i 


252 


CHAPTER  XIV 


of  the  principles  as  directly  applicable  to  the  characters 
of  manifestations  and  therefore  only  derivatively  to 
the  manifestations  themselves.  Hence  the  potential 
range  for  which  these  principles  hold  extends  beyond 
the  actually  existent  into  the  domain  of  the  possibly 
existent.  In  this  way  the  universality  of  law  is  wider 
than  that  of  fact.  While  the  universals  of  fact  are 
implied  by  the  universals  of  law,  the  statement  of  the 
latter  has  intrinsic  significance  not  involved  in  that  of 
the  former. 


INDEX 


Accordance,  grammatical  210 

„  with  fact  16,  17 

Adverbs,  as   secondary  adjectives 

102 
Alterable  predications  235,  242 

Analogy  207 
Analysis  107,  m,  112 
Analytical  proposition  63 
Applicatives  97 
Applied  mathematics  xxvi 
Art  of  thinking  xix 
Articles,  classified  85-88 
Assertion  as  conscious  belief  6 
Assertum  or  assertibile  4,  219 
Assurance  5 

Attainment  of  truth  xvii,  xviii 
Axioms  6  ^ 

Belief  5  ^ 

„    ,  degrees  of  5 
Bi-verbal  definitions  90 
Brackets  216,  217,  218 

Categories  and  parts  of  speech  9 
Causal  propositions  218 
Cause  and  effect  xxxvi,  242 

„  ,  as  reciprocal  242 

Certified  and  uncertified  55 
Characterisation  10 
Characterising  relation  213 
Characterising  tie  10 
Classes  121,  126 
Cognates  216,  217 
Commentary  propositions  166,  167 
Complementary    propositions     31, 

Comprehension  100 


Comprise  113 

Conceptualist  xxvii,  xxviii,  xxx 

Conjunctions  27 

Connectional  determination  237 

Connotation  100 
Consistency,  logic  of  222 
Constitutive  2 
Contingent  60 
Continuant  199 
Continuous  183 
Couples  205 

%,■:•  -    -       •-yp4^ 
Definition,  as  bi verbal  90 

„         ,  procedure  for  90,  104 

„         ,  purpose  of  90 
Demonstratives  88 
Determinates  174 
Determinandum  and   determinans 

9,  10 
Diadic  adjectives  203 
Diagrams:  Venn's  and  Euler's  124, 

125,  149-155 
Difference  and  otherness  xii 

Dilemma  37 

Discordance  with  fact  16,  17 
Disparate  190 
Distensive  magnitude  191 

Enumeration  and  number  114,  121 

„  of  items  113 

Enumerations  of  different  orders  117 

Enumerative  and  28,  122 
Epistemic  2 
Etymology  84 

Exclamatory  propositions  18 
Existence  of  individual  and  of  class 

171,  172 


Judgment  2 

Kant  on  Space  and  Time  24 


254  INDEX 

Existential  formulation  134,  160 
Experience  in  general  58 «. 
Experientially  certified  56 
Extension  124 

Fact  and  proposition  15,  62 
Factual  indication  92 
False  232 

Fictitious  narratives  168,  169 
Formal  logic  222 
Formally  certified  57 
Fundamentum  divisionis  173 

General  names  and  applicatives  97 
Grammar,  universal  xxi,  9 
Grammatical  analysis  8 
Governance,  grammatical  211 
Groups  28 

Hypothesis  6n. 
Hypothetical  attitude  44 

Ideas  164,  220 
Identity  and  otherness  186 
„       and  the  copula  13 
Images,  universe  of  162 
Imperatives  224 
Impersonal  propositions  19 
Impossibile,  reductio  ad  38 
Include  and  comprise  120,  121 
Indefinable  105 
Indemonstrable  223 
Independence  31,  137 
Induction  xiv,  xv,  xvi,  xxxv 
Inflections,  syntactic  and  significant 

211 
Instantial  160 

Intension  and  extension  124 
"Introduction"  and  proper  name 

84 
Items  113 


INDEX 


255 


Language  9 

Laws  of  nature  244 

„     of  thought  222,  224 
Leibniz  on  identity  of  indiscernibles 

194 

Material  logic  xxxiii 
Mathematics  xxiv,  xxv 
Modal  adjectives  52 
Modus  ponendo  ponens  etc.  34 

Narrative  propositions  166 
Necessary,  three  meanings  of  61 
Negation  v 
Nomic  61 
Nominalism  xxix 
Normative  xx,  xxi,  225 

Obversion  71,  73 
Occurrent   199 
Operators  114,  115,  116 
Ostensive  definition  94 
Otherness  and  difference  193 

Paradox  of  implication  39-47 

„        of  relations  211 
Particulars  or  individuals  1 1 

„         as  uncharacterised  12 
Partition  no 
Possibile  14,  217,  218 
Possible,  three  meanings  of  61 
Postulate  6«.,  248 
Potential  inference  42 
Pragmatists  235 
Prepositions  209 
Privative  predications  239 «. 
Probability  xi 

Problematic  or  uncertified  55 
Proposition  and  fact  15 
Pure  negation  66 

Quasi-substantives  215 

Ratio  208 
Realism  xxviii 


Referential  article  86 
Remainder  to  enumerations  113 

„  to  classes  146 

Resolution  and  complexes  1 10 

Secondary  propositions  iv,  166,  199 

„         adjectives  102 
Sense-experience  xvii 
Separate  presentment  of  sense-data 

20 
Standard  of  evaluation  226 
Subsistential  propositions  158,  159 
Suppositions  169 
Symmetrical  and  transitive  relations 

128,  129 
Synthetic  definitions  108,  109 

Thing  98,  99>  13° 
This  as  the  given  25 


Thought  including  perception  xvii 
Tie,  assertive  12 

„    ,  characterising  10 

„    ,  coupling  209 
Timelessness  of  truth  234,  235 
Transitive  adjectives  204«. 
Truisms  223 

Uniquely  descriptive  names  85 
Unit  adjective  52 
„    of  thought  10 
„    relation  213 
Universals  qui  adjectives  1 1 
Universe  of  descriptions  165 

of  discourse  161,  165 
of  ideas  163 
of  images  162 


w 


»» 


Verbal  propositions  62 


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LOGIC 


PART  11 


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CAMBRIDGE  UNIVERSITY  PRESS 

C.  F.  CLAY,  Manager 

LONDON   :  FETTER  LANE,  E.G.  4 


NEW  YORK  :  THE  MACMILLAN  CO. 
BOMBAY       \ 

CALCUTTA  I  MACMILLAN  AND  CO.,  Ltd. 
MADRAS      j 

TORONTO   :   THE  MACMILLAN  CO.  OF 
CANADA,  Ltd. 

TOKYO:  MARUZEN.KABUSHIKI-KAISHA 


ALL  RIGHTS  RESERVED 


1 


LOGIC 

FAR.T  .11 . 

t  >  I  <  1 1  « • 

DEMON  STRATI  V£  I'lSfF'fe'RENCE 
DEDUCTIVE  -AND-  INDUCTIVE 


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t  t  »      f  It* 


1  I 


BY 


W.  E.  JOHNSON,  M.A. 

FELLOW  OF  king's  COLLEGE,  CAMBRIDGE, 

SIDGWICK  LECTURER  IN  MORAL  SCIENCE  IN  THE 

UNIVERSITY  OF  CAMBRIDGE 


CAMBRIDGE 
AT  THE  UNIVERSITY  PRESS 

1922 


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2-2-  -  ?-  2,  ^  ^  ^ 


1/1 


CONTENTS 


i. 


INTRODUCTION 

PAGE 

Application  of  the  term  *  substantive ' xi 

Application  of  the  term  *  adjective ' xii 

Terms  '  substantive '  and  *  adjective '  contrasted  with  *  particular '  and 

*  universal '         . xiii 

Epistemic  character  of  assertive  tie     ......         .  xiv 

*  The  given '  presented  under  certain  determinables     ....  xiv 

The  paradox  of  implication xv 

Defence  of  Mill's  analysis  of  the  syllogism  ......  xvii 


CHAPTER  I 


^G3 


\k     \ 


INFERENCE  IN  GENERAL 

Implication  defined  as  potential  inference i 

I  Inferences  involved  in  the  processes  of  perception  and  association        .         2 

jConstitutive  and  epistemic  conditions  for  valid  inference.   Examination 
pf  the  *  paradox  of  inference '........         7 

'he  Applicative  and  Implicative  principles  of  inference      ...       10 

foint  employment  of  these  principles  in  the  syllogism  .         .        .if 

>istinction  between  applicational  and  implicational  universals.    The 
[structural  proposition  redundant  as  minor  premiss       .         .         .         .12 

>efinition  of  a  logical  category  in  terms  of  adjectival  determinables     .       15 

analysis  of  the  syllogism  in  terms  of  assigned  determinables.    Further 
lustrations  of  applicational  universals 17 

[ow  identity  may  be  said  to  be  involved  in  every  proposition     .         .       ao 

'he  formal  principle  of  inference  to  be  considered  redundant  as  major 
)remiss.  Illustrations  from  syllogism,  induction,  and  mathematical 
equality     .         .         .         .         .         .         .         .         .         .         ,         .20 

Mticism  of  the  alleged  subordination  of  induction  under  the  syllogistic 
)rinciple 24 


yi  CONTENTS  ! 

CHAPTER  II  { 

THE  RELATIONS  OF  SUB-ORDINATION  AND  CO-ORDIN 
TION  AMONGST  PROPOSITIONS  OF  DIFFERENT  TYPEi 


PA 


8  I    The  Counter-applicative  and  Counter-implicative  principles  required 

for  the  establishment  of  the  axioms  of  Logic  and  Mathematics  . 
§2.  Explanation  of  the  Counter-applicative  principle  .  .  .  . 
§3.  Explanation  of  the  Counter-implicative  principle  .  .  .  • 
§  4.  Significance  of  the  two  inverse  principles  in  the  philosophy  of  thought 
§  5.  Scheme  of  super-ordination,  sub-ordination  and  co-ordination  amongst 
propositions  ....•••••** 
§  6.  Further  elucidation  of  the  scheme 


\ 


§1. 
§3. 


§4. 
§5- 

§6. 
§7. 
§8. 
§9- 

§10. 

§11. 
§12. 

§13. 
§14- 
§15- 


CHAPTER  HI 

SYMBOLISM  AND  FUNCTIONS 

The  value  of  symbolism.  Illustrative  and  shorthand  symbols.  Classifi- 
cation of  formal  constants.  Their  distinction  from  material  constants  . 
The  nature  of  the  intelligence  required  in  the  construction  of  a  symbolic 

system 

The  range  of  variation  of  illustrative  symbols  restricted  within  some 
logical  category.  Combinations  of  such  symbols  further  to  be  inter- 
preted as  belonging  to  an  understood  logical  category.  Illustrations  of 
intelligence  required  in  working  a  symbolic  system  .  .  .  . 
Explanation  of  the  term  *  function,'  and  of  the  *  variants '  for  a  function 
Distinction  between  functions  for  which  all  the  material  constituents 
are  variable,  and  those  for  which  only  some  are  variable.   Illustrations 

from  logic  and  arithmetic 

The  various  kinds  of  elements  of  form  in  a  construct    .... 

Conjunctional  and  predicational  functions 

Connected  and  unconnected  sub-constructs 

The  use  of  apparent  variables  in  symbolism  for  the  representation  of  v'^e 
distributives  every  and  some.    Distinction  between  apparent  variables 

and  class-names 

Discussion  of  compound  symbols  which  do  and  which  do  not  represent 
genuine  constructs      ....•••••• 

Illustrations  of  genuine  and  fictitious  constructs 

Criticism  of  Mr  Russell's  view  of  the  relation  between  propositional 

functions  and  the  functions  of  mathematics 

Explanation  of  the  notion  of  a  descriptive  function  .... 
Further  criticism  of  Mr  Russell's  account  of  propositional  functions  . 
Functions  of  two  or  more  variants 


CONTENTS  vii 

CHAPTER  IV 
THE  CATEGORICAL  SYLLOGISM 

PAGE 

Technical  terminology  of  syllogism n^ 

Dubious  propositions  to  illustrate  syllogism -7 

Relation  of  syllogism  to  antilogism *,% 

4.  Dicta  for  the  first  three  figures  derived  from  a  single  antilogistic  dictum, 
showing  the  normal  functioning  of  each  figure 7A 

5.  Illustration  of  philosophical  arguments  expressed  in  syllogistic  form     .  81 

6.  Re-formulation  of  the  dicta  for  syllogisms  in  which  all  the  propositions 
are  general g^ 

The  propositions  of  restricted  and  unrestricted  form  in  each  figure       .  84 

Special  rules  and  valid  moods  for  the  first  three  figures        ...  85 

Special  rules  and  valid  moods  for  the  fourth  figure      ....  87 

Justification  for  the  inclusion  of  the  fourth  figure  in  logical  doctrine     .  88 

Proof  of  the  rules  necessary  for  rejecting  invalid  syllogisms ...  89 

Summary  of  above  rules ;  and  table  of  moods  unrejected  by  the  rules 

of  quality ^^ 

|3.   Rules  and  tables  of  unrejected  moods  for  each  figure  •         ...  93 

4.  Combination  of  the  direct  and  indirect  methods  of  establishing  the  valid 
moods  of  syllogism ^ 

5.  Diagram  representing  the  valid  moods  of  syllogism     ....  97 

6.  The  Sorites       ...........  07 

7.  Reduction  of  irregularly  formulated  arguments  to  syllogistic  form         .  98 

8.  Enthymemes ,qq 

9.  Importance  of  syllogism j©. 


CHAPTER  V 
FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM 


\i 


V 

[|  Deduction  goes  beyond  mere  subsumptive  inference,  when  the  major 
premiss  assumes  the  form  of  a  functional  equation.    Examples     .         .     103 
A  functional  equation  is  a  universal  proposition  of  the  second  order,  the 
functional  formula  constituting  a  Law  of  Co-variation.         .         .         .105 
The  solutions  of  mathematical  equations  which  yield  single-valued  func- 
1    \  tions  correspond  to  the  reversibility  of  cause  and  effect        .         .         .106 
\  ?  Ipignificance  of  the  number  of  variables  entering  into  a  functional  formula     108 
S^fexample  of  a  body  falling /«  z/ar«tf uq 

5  frhe  logical  characteristics  of  connectional  equations  illustrated  by  thermal 
)jJbid  economic  equilibria jj| 

The  method  of  Residues  is  based  on  reversibility  and  is  purely  deductive     1 16 

Reasons  why  the  above  method  has  been  falsely  termed  inductive        .     119 

eparation  of  the  subsumptive  from  the  functional  elements  in  these 
xtensions  of  syllogism u© 


! 


I 


VUl 


CONTENTS 


§1. 

§^. 
§3. 

§4- 
§5. 


§6. 

§7. 
§8. 

§9. 

§10. 

§  II. 
§  12. 

§13- 


CHAPTER  VI 

FUNCTIONAL  DEDUCTION 

In  the  deduction  of  mathematical  and  logical  formulae,  new  theorems 
are  established  for  the  different  species  of  a  genus,  which  do  not  hold      •- 
lor  the  genus     , 

Explanation  of  the  Aristotelean  tdiov ^ 

In  functional   deduction,   the   equational   formulae  are  non-Iimitine 
Elementary  examples ,         ^ 

The  range  of  universality  of  a  functional  formula  varies  with  the  number 
ot  mdependent  variables  involved.  Employment  of  brackets.  Impor- 
tance of  distinguishing  between  connected  and  disconnected  compounds 
The  functional  nature  of  the  formulae  of  algebra  accounts  for  the  possi- 
bility of  deducing  new  and  even  wider  formulae  from  previously  estab- 
lished and  narrower  formulae,  the  Applicative  Principle  alone  being 
employed  ....... 

Mathematical  Induction 

The  logic  of  mathematics  and  the  mathematics  of  logic 
Distinction  between  premathematical  and  mathematical  logic 
Formal  operators  and  formal  relations  represented  by  shorthand  and 
not  variable  symbols.    Classification  of  the  main  formal  relations  ac- 
cording to  their  properties 

The  material  variables  of  mathematical  and  logical  symbolisation  receive 

specihc  values  only  in  concrete  science 

Discussion  of  the  Principle  of  Abstraction  ....!* 
The  specific  kinds  of  magnitude  are  not  determinates  of  the  single  de- 
terminable Magnitude,  but  are  incomparable  ..... 
The  logical  symbolic  calculus  QsXMxshts  formulae  of  implication  vfhicYi 
are  to  be  contrasted  with  the  principles  of  inference  employed  in  the 
procedure  of  building  up  the  calculus  .  H    /        "  i"e 


CHAPTER  Vn 

THE  DIFFERENT  KINDS  OF  MAGNITUDE 

§  I.  The  terms  ^greater'  and  'less'  predicated  of  magnitude,  'larger'  and 

•smaller' ofthat  which  has  magnitude         .         .         . 
§  2.  Integral  number  as  predicable  of  classes  or  enumerations 
§  3.  Psychological  exposition  of  counting  .... 
§  4-  Logical  principles  underlying  counting 
§  5.  One-one  correlations  for  finite  integers 
§  6.   Definition  of  extensive  magnitude       .... 
§  7.  Adjectival  stretches  compared  with  substantival  . 
§  8.  Comparison  between  extensive  and  extensional  wholes 
§  9.  Discussion  of  distensive  magnitudes 

§  10.  Intensive  magnitude  . 

•         •         .         •         . 

§11.  Fundamental  distinction  between  distensive  and  intensive 


magni 


udes 


I. 
•I 


t] 


CONTENTS 


The  problem  of  equality  of  extensive  wholes 

Conterminus  spatial  and  temporal  wholes  to  be  considered  equal, 
tative  stretches  only  comparable  by  causes  or  effects    . 

Complex  magnitudes  derived  by  combination  of  simplex 

The  theory  of  algebraical  dimensions 


The  special  case  in  which  dividend  and  divisor  are  quantities 
same  kind  ......... 


Summary  of  the  above  treatment  of  magnitude 


quali 


of  the 


IX 

PAGE 
174 

180 

185 

186 
187 


CHAPTER  Vni 

INTUITIVE  INDUCTION 

The  general  antithesis  between  induction  and  deduction      .         .         .     1 89 

The  problem  of  abstraction 190 

The  principle  of  abstractive  or  intuitive  induction        .         .         .         .191 

Experiential  and  formal  types  of  intuitive  induction     .         .         .         .192 

Intuitive  induction  involved  in  introspective  and  ethical  judgments       .     193 

Intuitive  inductions  upon  sense-data  and  elementary  algebraical  and 
logical  relations .         .194 

Educational  importance  of  intuitive  induction 196 


CHAPTER  IX 

SUMMARY  INCLUDING  GEOMETRICAL  INDUCTION 

Summary  induction  reduced  to  first  figure  syllogism     .         .         .         .197 

Summary  induction  as  establishing  the  premiss  for  induction  proper. 

Criticism  of  Mill's  and  Whewell's  views 198 

Summary  induction  involved  in  geometrical  proofs      ....  200 
I  Explanation  of  the  above  process        ,         .        .         .         .         .         .201 

Function  of  the  figure  in  geometrical  proofs 203 

Abuse  of  the  figure  in  geometrical  proofs 205 

Criticism  of  Mill's 'parity  of  reasoning' 208 


CHAPTER  X 

DEMONSTRATIVE  INDUCTION 

\  ^^Demonstrative  induction  uses  a  composite  along  with  an  instantial 

premiss 210 

Illustrations  of  demonstrative  arguments  leading  up  to  demonstrative 
induction 210 

Conclusions  reached  by  the  conjunction  of  an  alternative  with  a  dis- 
junctive premiss         .        .        - 214 

«5 


;l  ,  ' 


/ 


CONTENTS 


PA 


§  4.  The  formula  of  direct  universalisation 1 

§  5.  Scientific  illustration  of  the  above V 

§  6.   Proposed  modification  of  Mill's  exposition  of  the  methods  of  induction  ^ 

§  7.  The  major  premiss  for  demonstrative  induction  as  an  expression  of  the  ^^  rT^>r-v     -r-iA-nk'-n     tt 

dependence  in  the  variations  of  one  phenomenal  character  upon  those  ^H  INTRODUCTION    TO     PAR  1      11 

of  others 

§  8.  The  four  figures  of  demonstrative  induction ^^  ,  .        , 

§  9.  Figure  of  Difference B         §i.    Before  introducing  the  topics  to  be  examined 

§10.  Figure  of  Agreement ■  in  Part  II    I  oropose  to  recapitulate  the  substance  of 

§11.  Figure  of  Composition ^H  '  ,    .  1     .  •  ^'  '*.U 

§1..  Figure  of  Resolution ■   Part  I,  and  in  so  doing  to  bring  into  connection  with 

§  13.  The  Antiiogism  of  Demonstrative  Induction ^1  ^^^  another  Certain  problettis  which  Were  there  treated 

§14.   Illustration  of  the  Figure  of  Difference ^H  t    t_  ^U         4.^  1^,,  ^:fr^,-^«4-  c^m 

i.t  Illustration  of  the  Fi^re  of  Agreement ■  in  different  chapters.    I  hope  thus  to  lay  different  em- 

§16.  Principle  for  dealing  with  cases  in  which  a  number  both  of  cause-factors     ^M    pJ^^Lsis  UDOn  SOme  of  the  theories  that  have  been  main- 
and  effect-factors  are  considered,  with  a  symbolic  example  .         .         .     ^M    r  ir  .  4-       A'    rrc 

§17.  Modification  of  symbolic  notation  in  the  figures  where  different  cause-  g   tained,  and  to  remove  any  possible  misunderstanQings 

factors  represent  determinates  under  the  same  determinable  .         .    H     ^Uf^^f^  the  treatment  waS  Unavoidably  COndensed. 

§  18.  The  striking  distinction  between  the  two  last  and  the  two  first  figures .    W     Where  tne  treacmeni  wab  uiidvuiu         y 
§  19.  Explanation  of  the  distinction  between  composition  and  combination    |H  In  my  analysis of  the  prOpOSltlOn  1  havedlStingUlSnea 

of  cause-factors  .      .  '^,'      '      *      '    M    the  natures  of  substantive  and  adjective  in  a  form  in- 

§20.  Illustrations  of  the  figures  of  Composition  and  Resolution   .         .         .    j^     tne   naiurcb   ui    auustdiiui  vv.  j  ^ 

tended  to  accord  in  essentials  with  the  doctrine  ot  the 
CHAPTER  XI  ■    large  majority  of  logicians,  and  as  far  as  my  terminology 

is  new  its  novelty  consists  in  giving  wider  scope  to  each 
THE  FUNCTIONAL  EXTENSION^OF  DEMONSTRATIVES    ^^  ^^^^^  ^^^  fundamental  terms.    Prima  facie  it  might 

be  supposed  that  the  connection  of  substantive  with 
^^'  }^:t^\^:^Z:Zi:^^^^^  ■    adjective  in  the  construction  of  a  proposition  is  tanta- 

§  •2.  Contrast  between  my  exposition  and  Mill's ^|    mount  to  the  metaphysical  notions  of  substatice  and 

§3.  The  different  uses  of  the  term' hypothesis -iniogic    .       .       .      .   H    :„herence      But  HIV  notloii  of  substantive  is  intended 

§4.  Jevons's   confusion   between   the   notions   *. problematic'  and    'hypo-    ^H      innerencc.       UUL    iiiy    iivjliwii    yj*^    o« 

theticai' jH   to  include,  besides  the  metaphysical  notion  of  substance 

§  5.  The  establishment  of  a  functional  formula  for  the  figures  of  Difference    ■■  1  i-i  U'««lUr  ;,,o<-;^^r1       fli^  no 

and  of  Composition U    —SO  far  as  this  Can  be  philosophically  justitied— the  no- 

§6.  The  criteria  of  simplicity  and  analogy  for  selection  of  the  functional     ||1      ^.^^  of  OCCUrrenceS  Or  eventS  tO  which  SOme  philoSOpherS 
§7.  A   comparison   of   these   criteria  with  similar  criteria  proposed   by    ^Uj      of  the  present  day  wish  tO  restrict  the  realm  OI   reality. 

§8.  ^ciJcarit^^^^^^        m'ethods  for* deteimining  the  mo'st  probabk  «     Thus  by  a  substantive /r^/^r  I  mean  an  existent;  and 

formula !■    the  Category  of  the  existent  is  divided  into  the  two 

INDEX .      •      •    fl    subcategories:  what  continues  to  exist,  or  the  continuant; 

and  what  ceases  to  exist,  or  the  occurrent,  every  occur- 
rent  being  referrible  to  a  continuant.    To  exist  is  to  be 


xu 


INTRODUCTION 


INTRODUCTION 


xiu 


in  temporal  or  spatio-temporal  relations  to  other  exis- 
tents;  and  these  relations  between  existents  are  the 
fundamentally  external  relations.  A  substantive  proper 
cannot  characterise,  but  is  necessarily  characterised ;  on 
the  other  hand,  entities  belonging  to  any  category 
whatever  (substantive  proper,  adjective,  proposition, 
etc.)  may  be  characterised  by  adjectives  or  relations 
belonging  to  a  special  adjectival  sub-category  corre- 
sponding, in  each  case,  to  the  category  of  the  object 
which  it  characterises.  Entities,  other  than  substantives 
proper,  of  which  appropriate  adjectives  can  be  predi- 
cated, function  as  quasi-substantives. 

§  2.    The  term  adjective,  in  my  application,  covers 
a  wider  range  than  usual,  for  it  is  essential  to  my  system 
that  it  should  include  relations.    There  are  two  distinct 
points  of  view  from  which  the  treatment  of  a  relation 
as  of  the  same  logical  nature  as  an  adjective  may  be 
defended.    In  the  first  place  the  complete  predicate  in 
a  relational  proposition  is,  in  my  view,  relatively  to  the 
subject  of  such  proposition,  equivalent  to  an  adjective 
in  the  ordinary  sense.    For  example,  in  the  proposition, 
'  He  is  afraid  of  ghosts,'  the  relational  component  is  ex- 
pressed by  the  phrase  'afraid  of  ;    but  the  complete 
predicate  *  afraid  of  ghosts'  (which  includes  this  relation) 
has  all  the  logical  properties  of  an  ordinary  adjective, 
so  that  for  logical  purposes  there  is  no  fundamental  dis- 
tinction between  such  a  relational  predicate  and  an  irra- 
tional predicate.    In  the  second  place,  if  the  relational 
component  in  such  a  proposition  is  separated,  I  hold  that 
it  can  be  treated  as  an  adjective  predicated  of  the  sub- 
stantive-couple 'he'  and  'ghosts'.  In  other  words,  a  rela- 
tion cannot  be  identified  with  a  c/ass  of  couples,  i.e.  be 


conceived  extensionally  ;  but  must  be  understood  to 
characterise  couples,  i.e.  be  conceived  intensionally.  It 
seems  to  me  to  raise  no  controvertible  problem  thus  to 
include  relations  under  the  wide  genus  adjectives.  It  is 
compatible,  for  example,  with  almost  the  whole  of  Mr 
Russell's  treatment  of  the  proposition  in  his  Principles  of 
Mathematics]  and,  without  necessarily  entering  into  the 
controvertible  issues  that  emerge  in  such  philosophical 
discussions,  I  hold  that  some  preliminary  account  of 
relations  is  required  even  in  elementary  logic. 

§  3.    My  distinction  between  substantive  and  adjec- 
tive is  roughly  equivalent  to  the  more  popular  philoso- 
phical antithesis  between  particular  and  universal ;  the 
notions,  however,  do    not   exactly  coincide.     Thus   I 
understand  the  philosophical  term  particular  not  to  apply 
to  quasi-substantives,  but  to  be  restricted  to  substantives 
proper,  i.e.  existents,  or  even  more  narrowly  to  occur- 
rents.    On  the  other  hand,  I  find  a  fairly  unanimous 
opinion  in  favour  of  calling  an  adjective  predicated  of 
a  particular  subject,  a  particular— the  name  universal 
being  confined  to  the  abstract  conception  of  the  adjec- 
tive.   Thus  red  or  redness,  abstracted  from  any  specific 
judgment,  is  held  to  be  universal;    but  the  redness, 
manifested  in  a  particular  object  of  perception,  to  be 
itself  particular.    Furthermore,  qua  particular,  the  ad- 
jective is  said  to  be  an  existent,  apparently  in  the  same 
sense  as  the  object  presented  to  perception  is  an  exis- 
tent.   To  me  it  is  difficult  to  argue  this  matter  because, 
while  acknowledging  that  an  adjective  may  be  called  a 
universal,  I  regard  it  not  as  a  mere  abstraction,  but  as 
a  factor  in  the  real ;  and  hence,  in  holding  that  the  ob- 
jectively real  is  properly  construed  into  an  adjective 


XIV 


INTRODUCTION 


INTRODUCTION 


XV 


characterising  a  substantive,  the  antithesis  between  the 
particular  and  the  universal  (i.e.  in  my  terminology 
between  the  substantive  and  the  adjective)  does  not 
involve  separation  within  the  real,  but  solely  a  separation 
for  thought,  in  the  sense  that  the  conception  of  the 
substantive  apart  from  the  adjective,  as  well  as  the 
conception  of  the  adjective  apart  from  the  substantive, 
equally  entail  abstraction. 

§  4.    Again,  taking  the  whole  proposition  constituted 
by  the  connecting  of  substantive  with  adjective,  I  have 
maintained  that  in  a  virtually  similar  sense  the  proposi- 
tion is  to  be  conceived  as  abstract.    But,  whereas  the 
characterising  tie  may  be  called  constitutive  in  its  func- 
tion of  connecting  substantive  with  adjective  to  con- 
struct the  proposition,  I  have  spoken  of  the  assertive 
tie  as  epistemic,  in  the  sense  that  it  connects  the  thinker 
with  the  proposition  in  constituting  the  unity  which  may 
be  called  an  act  of  judgment  or  of  assertion.   When, 
however,  this  act  of  assertion  becomes  in  its  turn  an 
object  of  thought,  it  is  conceived  under  the  category  of 
the  existent ;  for  such  an  act  has  temporal  relations  to 
other  existents,  and  is  necessarily  referrible  to  a  thinker 
conceived  as  a  continuant.    Though,  relatively  to  the 
primary  proposition,  the  assertive  tie  must  be  conceived 
as  epistemic ;  yet,  relatively  to  the  secondary  proposition 
which  predicates  of  the  primary  that  it  has  been  asserted 
by  A,  the  assertive  tie  functions  constitutively. 

§  5.  In  view  of  a  certain  logical  condition  presup- 
posed throughout  this  Part  of  my  work,  I  wish  to  re- 
mind the  reader  of  that  aspect  of  my  analysis  of  the 
proposition,  according  to  which  I  regard  the  subject  as 
that  which  is  given  to  be  determinately  characterised 


by  thought.  Now  I  hold  that  for  a  subject  to  be 
characterised  by  some  adjectival  determinate,  it  must 
first  have  been  presented  as  characterised  by  the  corre- 
sponding adjectival  determinable.  The  fact  that  what 
is  given  is  characterised  by  an  adjectival  determinable 
is  constitutive ;  but  the  fact  that  it  is  presented  as  thus 
characterised  is  epistemic.  Thus,  for  a  surface  to  be 
characterised  as  red  or  as  square,  it  must  first  have 
been  constructed  in  thought  as  being  the  kind  of  thing 
that  has  colour  or  shape;  for  an  experience  to  be 
characterised  as  pleasant  or  unpleasant,  it  must  first 
have  been  constructed  in  thought  as  the  kind  of  thing 
that  has  hedonic  tone.  Actually  what  is  given,  is  to  be 
determined  with  respect  to  a  conjunction  of  several 
specific  aspects  or  determinables ;  and  these  determine 
the  category  to  which  '  the  given '  belongs.  For  example, 
on  the  dualistic  view  of  reality,  the  physical  has  to  be 
determined  under  spatio-temporal  determinables,  and 
the  psychical  under  the  determinable  consciousness  or 
experience.  If  the  same  being  can  be  characterised  as 
two-legged  and  as  rational,  he  must  be  put  into  the 
category  of  the  physico-psychical. 

§  6.  The  passage  from  topics  treated  in  Part  I  to 
those  in  Part  II,  is  equivalent  to  the  step  from  implica- 
tion to  inference.  The  term  inference,  as  introduced  in 
Part  I,  did  not  require  technical  definition  or  analysis, 
as  it  was  sufficiently  well  understood  without  explana- 
tion. It  was,  however,  necessary  in  Chapter  III  to  in- 
dicate in  outline  one  technical  difficulty  connected  with 
the  paradox  of  implication;  and  there  I  first  hinted, 
what  will  be  comprehensively  discussed  in  the  first 
chapter  of  this  Part,  that  implication  is  best  conceived 


XVI 


INTRODUCTION 


INTRODUCTION 


xvu 


■n 

Hi 


it 


\ 


as  potential  inference.    While  for  elementary  purposes 
implication  and  inference  may  be  regarded  as  practically 
equivalent,  it  was  pointed  out  in  Chapter  1 1 1  that  there 
is  nevertheless  one  type  of  limiting  condition  upon  which 
depends  the  possibility  of  using  the  relation  of  implica- 
tion for  the  purposes  of  inference.    Thus  reference  to 
the  specific  problem  of  the  paradox  of  implication  was 
unavoidable  in  Part  I,  inasmuch  as  a  comprehensive 
account  of  symbolic  and  mechanical  processes  necessarily 
included  reference  to  all  possible  limiting  cases;  but, 
apart  from  such  a  purely  abstract  treatment,  no  special 
logical  importance  was  attached  to  the  paradox.    The 
limiting  case  referred  to  was  that  of  the  permissible  em- 
ployment of  the  compound  proposition  'Up  then  ^,'in  the 
unusual  circumstance  where  knowledge  of  the  truth  or 
the  falsity  of/  or  of  ^  was  already  present  when  the  com- 
pound proposition  was  asserted.    This  limiting  case  will 
iftt  recur  in  the  more  important  developments  of  infer- 
ence that  will  be  treated  in  the  present  part  of  my  logic. 
It  might  have  conduced  to  greater  clearness  if,  in 
Chapters  III  and  IV,  I  had  distinguished— when  using 
the  phrase  implicative  proposition— h^X^^^^Vi  the  primary 
and  secondary  interpretations  of  this  form  of  proposi- 
tion.   Thus,  when  the  compound  proposition  *If/  then 
q'  is  rendered,  as  Mr  Russell  proposes,  in  the  form 
*  Either  not-/  or  q'  the  compound  is  being  treated  as  a 
primary  proposition  of  the  same  type  as  its  components 
p  and  q.  When  on  the  other  hand  we  substitute  for  '  If 
/  then  q  the  phrase  'p  implies  q;  or  preferably  '/  would 
imply  ql  the  proposition  is  no  longer  primary,  inasmuch 
as  it  predicates  about  the  proposition  q  the  adjective 
'implied  by  /'  which  renders  the  compound  a  secondary 


proposition,  in  the  sense  explained  in  Chapter  IV\  Now 
whichever  of  these  two  interpretations  is  adopted,  the 
inference  which  is  legitimate  under  certain  limiting  con- 
ditions is  the  same.    Thus  given  the  compound  *  Either 
not-/  or  q'  conjoined  with  the  assertion  of  */,'  we  could 
infer  *y';  just  as  given  'p  implies  q'  conjoined  with  the 
assertion  of  */,'  we  infer  'q!    It  is  for  this  reason  that 
the  two  interpretations  !iave  become  merged  into  one 
in  the  ordinary  symbolic  treatment  of  compound  pro- 
positions; and  in  normal  cases  no  distinction  is  made 
in  regard  to  the  possibility  of  using  the  primary  or 
secondary  interpretation  for  purposes  of  inference.    The 
normal  case,  however,  presupposes  that  /  and  q  are 
entertained  hypothetically ;  when  this  does  not  obtain, 
the  danger  of  petitio  principii  enters.   The  problem  in 
Part  I  was  only  a  very  special  and  technical  case  in 
which  this  fallacy  has  to  be  guarded  against ;  in  Part  II, 
it  will  be  dealt  with  in  its  more  concrete  and  philoso- 
phically important  applications. 

§  7.  The  mention  of  this  fallacy  immediately  sug- 
gests Mill's  treatment  of  the  functions  and  value  of  the 
syllogism;  but,  before  discussing  his  views,  I  propose 
to  consider  what  his  main  purpose  was  in  tackling  the 
charge  of  petitio  principii  that  had  been  brought  against 
the  whole  of  formal  argument,  including  in  particular 
the  syllogism.  In  the  first  section  of  his  chapter.  Mill 
refers  to  two  opposed  classes  of  philosophers — the  one 
of  whom  regarded  syllogism  as  the  universal  type  of  all 
logical  reasoning,  the  other  of  whom  regarded  syllogism 

^  The  interpretation  of  the  implicative  form  '/  implies  q'  as 
secondary  is  developed  in  Chapter  III,  §  9,  where  the  modal  adjectives 
necessary^  possible^  impossible^  are  introduced. 


I 


xvm 


INTRODUCTION 


INTRODUCTION 


XIX 


I'll 


III 


as  useless  on  the  ground  that  all  such  forms  of  inference 
xnwoXv^ petitio principil    He  then  proceeds:  ^  believe 
both  these  opinions  to  be  fundamentally  erroneous,' and 
this  would  seem  to  imply  that  he  proposed  to  relieve 
the  syllogism  from  the  charge.    I  believe,  however,  that 
all  logicians  who  have  referred  to    Mill's  theory— a 
group  which  includes  almost  everyone  who  has  written 
on  the  subject  since  his  time— have  assumed  that  the 
purport  of  the  chapter  was  to  maintain  the  charge  of 
petitio  principii,  an  interpretation  which  his  opening 
reference  to  previous  logicians  would  certainly  not  seem 
to  bear.    His  subsequent  discussion  of  the  subject  is, 
verbally  at  least,  undoubtedly  confusing,  if  not  self-con- 
tradictory ;  but  my  personal  attitude  is  that,  whatever 
may  have  been  Mill's  general  purpose,  it  is  from  his  own 
exposition  that  I,  in  common  with  almost  all  his  con- 
temporaries, have  been  led  to  discover  the  principle 
according  to  which  the  syllogism  can  be  relieved  from 
the  incubus  to  which  it  had  been  subject  since  the  time 
of  Aristotle.    In  my  view,  therefore,  Mill's  account  of 
the  philosophical  character  of  the  syllogism  is  incon- 
trovertible ;  I  would  only  ask  readers  to  disregard  from 
the  outset  any  passage  in  his  chapter  in  which  he 
appears  to  be  contending  for  the  annihilation  of  the 
syllogism  as  expressive  of  any  actual  mode  of  inference. 
Briefly  his  position  may  be  thus  epitomised.   Taking 
a  typical  syllogism  with  the  familiar  major  ^All  men 
are  mortal,'  he  substituted  for  ^Socrates'  or  *  Plato'  the 
minor  term  *the  Duke  of  Wellington'  who  was  then 
living.    He  then  maintained  that,  going  behind  the 
syllogism,  certain  instantial  evidence  is  required  for  es- 
tablishing the  major;  and  furthermore  that  the  validity 


>f  the  conclusion  that  the  Duke  of  Wellington  would 

lie  depends  ultimately  on  this  instantial  evidence.   The 

iterpolation  of  the  universal  major  *  All  men  will  die ' 

las  undoubted  value,  to  which  Mill  on  the  whole  did 

istice;  but  he  pointed  out  that  the  formulation  of  this 
iniversal  adds  nothing  to  the  positive  or  factual  data 
pon  which  the  conclusion  depends.  It  follows  from 
lis  exposition  that  a  syllogism  whose  major  is  admittedly 
[stablished  by  induction  from  instances  can  be  relieved 
rom  the  reproach  of  begging  the  question  or  circularity 
If,  and  only  if,  the  minor  term  is  not  included  in  the 
lltimate  evidential  data.  The  Duke  of  Wellington  being 
[till  living  could  not  have  formed  part  of  the  evidence 
|pon  which  the  universal  major  depended.   It  was  there- 

)re  part  of  Mill's  logical  standpoint  to  maintain  that 

lere  were  principles  of  induction  by  which,  from  a 
Imited  number  of  instances,  a  universal  going  beyond 

lese  could  be  logically  justified.  This  contention  may 
)e  said  to  confer  constitutive  validity  upon  the  inductive 
process.  It  is  directly  associated  with  the  further  con- 
[ideration  that  an  instance,  not  previously  examined,  may 
)e  adduced  to  serve  as  minor  premiss  for  a  syllogism, 
md  that  such  an  instance  will  always  preclude  circularity 
in  the  formal  process.  Now  the  charge  of  circularity  or 
)etitio  principii  is  epistemic;  and  the  whole  of  Mill's 
irgument  may  therefore  be  summed  up  in  the  statement 
that  the  epistemic  validity  of  syllogism  and  the  consti- 
tutive validity  of  induction,  both  of  which  had  been  dis- 

)uted  by  earlier  logicians,  stand  or  fall  together. 

In  order  to  prevent  misapprehension  in  regard  to 
jU's  view  of  the  syllogism,  it  must  be  pointed  out  that 

le  virtually  limited  the  topic  of  his  chapter  to  cases  in 


XX 


INTRODUCTION 


which  the  major  premiss  would  be  admitted  by  all 
logicians  to  have  been  established  by  means  of  induction 
in  the  ordinary  sense,  i.e.  by  the  simple  enumeration  of 
instances;  although  many  of  them  would  have  contended 
that  such  instantial  evidence  was  not  by  itself  sufficient. 
Thus  all  those  cases  in  which  the  major  was  otherwise 
established,  such  as  those  based  on  authority,  intuition 
or  demonstration,  do  not  fall  within  the  scope  of  Mill's 
solution.    Unfortunately  all  the  commentators  of  Mill 
have  confused  his  view  that  universals  cannot  be  in- 
tuitively but  only  empirically  established,  with  his  spe- 
cific contention  in  Chapter  IV.    I  admit  that  he  himself 
is  largely  responsible  for  this  confusion,  and  therefore, 
while  supporting  his  view  on  the  functions  of  the  syl- 
logism, I  must  deliberately  express  my  opposition  to 
his  doctrine  that  universalis  can  only  ultimately  be  estab- 
lished empirically,  and  limit  my  defence  to  his  analysis 
of  those  syllogisms  in  which  it  is  acknowledged  that  the 
major  is  thus  established.    Even  here  his  doctrine  that 
all  inference  is  from  particulars  to  particulars  is  open  to 
fundamental  criticism  ;    and,  in   my  treatment  of  the 
principles  of  inductive  inference  which  will  be  developed 
in  Part  III,  I  shall  substitute  an  analysis  which  will 
take  account  of  such  objections  as  have  been  righdy 
urged  against  Mill's  exposition. 

[Note.  There  are  two  cases  in  which  the  technical  terminology 
employed  in  Part  II  differs  from  that  in  Part  I.  (i)  The  phrase/r/W- 
tive  proposition,  in  Part  I,  is  to  be  understood  psychologically;  in 
Part  II,  logically  as  equivalent  to  axiom,  (2)  Counter-implicative,  in 
Part  I,  applies  to  the  form  of  a  compound  proposition;  in  Part  II,  to 
a  principle  of  inference.] 


CHAPTER  I 

INFERENCE  IN  GENERAL 

§  I.  Inference  is  a  mental  process  which,  as  such, 
las  to  be  contrasted  with  implication.  The  connection 
)etween  the  mental  act  of  inference  and  the  relation 
)f  implication  is  analogous  to  that  between  assertion  and 
the  proposition.  Just  as  a  proposition  is  what  is  poten- 
tially assertible,  so  the  relation  of  implication  between 
:wo  propositions  is  an  essential  condition  for  the  possi- 
Ibility  of  inferring  one  from  the  other;  and,  as  it  is 
'impossible  to  define  a  proposition  ultimately  except  in 
terms  of  the  notion  of  asserting,  so  the  relation  of  im- 
plication can  only  be  defined  in  terms  of  inference. 
This  consideration  explains  the  importance  which  I 
attach  to  the  recognition  of  the  mental  attitude  involved 
in  inference  and  assertion ;  afterwhich  the  stricdy  logical 
question  as  to  the  distinction  between  valid  and  invalid 
inference  can  be  discussed.  To  distinguish  the  formula 
I  of  implication  from  that  of  inference,  the  former  may 
|be  symbolised  '\i p  then  q^  and  the  latter  'p  therefore 
,'  where  the  symbol  q  stands  for  the  conclusion  and/ 
^or  the  premiss  or  conjunction  of  premisses. 

The  proposition  or  propositions  from  which  an  in- 
perence  is  made  being  called  premisses,  and  the  pro- 
)osition  inferred  being  called  the  conclusion,  it  is 
:ommonly  supposed  that  the  premisses  are  the  pro- 
)ositions  first  presented  in  thought,  and  that  the  transi- 
tion from  these  to  the  thought  of  the  conclusion  is  the 
J.  L.  n  , 


XX 


INTRODUCTION 


which  the  major  premiss  would  be  admitted  by  all 
logicians  to  have  been  established  by  means  of  induction 
in  the  ordinary  sense,  i.e.  by  the  simple  enumeration  of 
instances;  although  many  of  them  would  have  contended 
that  such  instantial  evidence  was  not  by  itself  sufficient. 
Thus  all  those  cases  in  which  the  major  was  otherwise 
established,  such  as  those  based  on  authority,  intuition 
or  demonstration,  do  not  fall  within  the  scope  of  Mills 
solution.  Unfortunately  all  the  commentators  of  Mill 
have  confused  his  view  that  universals  cannot  be  in- 
tuitively but  only  empirically  established,  with  his  spe- 
cific contention  in  Chapter  IV.  I  admit  that  he  himself 
is  largely  responsible  for  this  confusion,  and  therefore, 
while  supporting  his  view  on  the  functions  of  the  syl- 
logism, I  must  deliberately  express  my  opposition  to 
his  doctrine  that  universals  can  only  ultimately  be  estab- 
lished empirically,  and  limit  my  defence  to  his  analysis 
of  those  syllogisms  in  which  it  is  acknowledged  that  the 
major  is  thus  established.  Even  here  his  doctrine  that 
all  inference  is  from  particulars  to  particulars  is  open  to 
fundamental  criticism  ;  and,  in  my  treatment  of  the 
principles  of  inductive  inference  which  will  be  developed 
in  Part  III,  I  shall  substitute  an  analysis  which  will 
take  account  of  such  objections  as  have  been  righdy 
urged  against  Mills  exposition. 

[Note.  There  are  two  cases  in  which  the  technical  terminology 
employed  in  Part  II  differs  from  that  in  Part  I.  (i)  The  phrase /n;«/- 
tive  proposition^  in  Part  I,  is  to  be  understood  psychologically;  in 
Part  II,  logically  as  equivalent  to  axiom,  (2)  Counier-implicaiive,  in 
Part  I,  applies  to  the  form  of  a  compound  proposition;  in  Part  II,  to 
a  principle  of  inference.] 


CHAPTER  I 

INFERENCE  IN  GENERAL 

§  I.    Inference  is  a  mental  process  which,  as  such, 

has  to  be  contrasted  with  implication.    The  connection 

between  the  mental  act  of  inference  and  the  relation 

j  of  implication  is  analogous  to  that  between  assertion  and 

the  proposition.   Just  as  a  proposition  is  what  is  poten- 

'tially  assertible,  so  the  relation  of  implication  between 

I  two  propositions  is  an  essential  condition  for  the  possi- 

jbility  of  inferring  one  from  the  other;  and,  as  it  is 

impossible  to  define  a  proposition  ultimately  except  in 

[terms  of  the  notion  of  asserting,  so  the  relation  of  im- 

jplication  can  only  be  defined   in  terms  of  inference. 

This  consideration   explains  the  importance  which  I 

attach  to  the  recognition  of  the  mental  attitude  involved 

in  inference  and  assertion ;  afterwhich  the  stricdy  logical 

question  as  to  the  distinction  between  valid  and  invalid 

jinference  can  be  discussed.    To  distinguish  the  formula 

lof  implication  from  that  of  inference,  the  former  may 

be  symbolised  'lip  then  q^  and  the  latter  'p  therefore 

1^,'  where  the  symbol  q  stands  for  the  conclusion  and/ 

[for  the  premiss  or  conjunction  of  premisses. 

The  proposition  or  propositions  from  which  an  in- 
ference is  made  being  called  premisses,  and  the  pro- 
)osition  inferred  being  called  the  conclusion,  it  is 
:ommonly  supposed  that  the  premisses  are  the  pro- 
>ositions  first  presented  in  thought,  and  that  the  transi- 
tion from  these  to  the  thought  of  the  conclusion  is  the 

J.  L.  II  . 


V 

j 


2  CHAPTER  I 

last  step  in  the  process.  But  in  fact  the  reverse  is 
usually  the  case;  that  is  to  say,  we  first  entertain  in 
thought  the  proposition  that  is  technically  called  the 
conclusion,  and  then  proceed  to  seek  for  other  pro- 
positions which  would  justify  us  in  asserting  it.  The 
conclusion  may,  on  the  one  hand,  first  present  itself  to 
us  as  potentially  assertible,  in  which  case  the  mental 
process  of  inference  consists  in  transforming  what  was 
potentially  assertible  into  a  proposition  actually  asserted. 
On  the  other  hand,  we  may  have  already  satisfied 
ourselves  that  the  conclusion  can  be  validly  asserted 
apart  from  the  particular  inferential  process,  in  which 
case  we  may  yet  seek  for  other  propositions  which, 
functioning  as  premisses,  would  give  an  independent  or 
additional  justification  for  our  original  assertion.  In 
every  case,  the  process  of  inference  involves  three  dis- 
tinct assertions :  first  the  assertion  of  */,'  next  the  asser- 
tion of  *^/  and  thirdly  the  assertion  that  'p  would  imply 
q'  It  must  be  noted  that  'p  would  imply  ^,'  which  is 
the  proper  equivalent  of  'if/  then  q,'  is  the  more  correct 
expression  for  the  relation  of  implication,  and  not  'p 
implies  q' — which  rather  expresses  the  completed  in- 
ference. This  shows  that  inference  cannot  be  defined 
in  terms  of  implication,  but  that  implication  must  be 
defined  in  terms  of  inference,  namely  as  equivalent  to 
potential  inference.  Thus,  in  inferring,  we  are  not 
merely  passing  from  the  assertion  of  the  premiss  to  the 
assertion  of  the  conclusion,  but  we  are  also  implicitly 
asserting  that  the  assertion  of  the  premiss  is  used  to 
justify  the  assertion  of  the  conclusion. 

§  2.    Some  difficult  problems,  which  are  of  special 
importance  in  psychology,  arise  in  determining  quite 


INFERENCE  IN  GENERAL  3 

precisely  the  range  of  those  mental  processes  which 
may  be  called  inference',  in  particular,  how  far  asser- 
tion or  inference  is  involved  in  the  processes  of  asso- 
ciation and  of  perception.  These  difficulties  have  been 
aggravated  rather  than  removed  by  the  quite  false 
antithesis  which  some  logicians  have  drawn  between 
logical  and  psychological  inference.  Every  inference  is 
a  mental  process,  and  therefore  a  proper  topic  for  psycho- 
logical analysis ;  on  the  other  hand,  to  infer  is  to  think, 
and  to  think  is  virtually  to  adopt  a  logical  attitude;  for 
everyone  who  infers,  who  asserts,  who  thinks,  intends 
to  assert  truly  and  to  infer  validly,  and  this  is  what  con- 
stitutes assertion  or  inference  into  a  logical  process.  It 
is  the  concern  of  the  science  of  logic,  as  contrasted  with 
psychology,  to  criticise  such  assertions  and  inferences 
from  the  point  of  view  of  their  validity  or  invalidity. 

Let  us  then  consider  certain  mental  processes — in 
particular  processes  of  association — which  have  the 
semblance  of  inference.  In  the  first  place,  there  are 
many  unmistakeable  cases  of  association  in  which  no 
inference  whatever  is  even  apparently  involved.  Any 
familiar  illustration,  either  of  contiguity  or  of  similarity, 
will  prove  that  association  in  itself  does  not  entail  in- 
ference. If  a  cloudy  sky  raises  memory-images  of  a 
storm,  or  leads  to  the  mental  rehearsal  of  a  poem,  or 
suggests  the  appearance  of  a  slate  roof,  in  none  of  these 
revivals  by  association  is  there  involved  anything  in  the 
remotest  degree  resembling  inference.  The  case  of  con- 
tiguity is  that  which  is  most  commonly  supposed  to 
involve  some  sort  of  inference;  but  in  this  supposal  there 
is  a  confusion  between  recollection  and  expectation. 
Our  recollection  of  storms  that  we  have  experienced  in 


4  CHAPTER  I 

the  past  is  obviously  distinct  from  our  expectation  that 
a  storm  is  coming  on  in  the  immediate  future.    It  is  to 
this  latter  process  of  expectation,  and  not  to  the  former 
process  of  recollection,  that  the  term  inference  is  more 
or  less  properly  applied ;  but  even  here  we  must  make 
a  careful  psychological  distinction.    We  may  expect  a 
storm  when  we  notice  the  darkness  of  the  sky,  without 
at  all  having  actually  recalled  past  experiences  of  storms; 
in  this  case  no  inference  is  involved,  since  there  has 
been  only  one  assertion,  namely,  what  would  constitute 
the  conclusion  without  any  other  assertion  that  would 
constitute  a  premiss.    In  order  to  speak  properly  of 
inference  in  such  cases,  the  minimum  required  is  the 
assertion  that  the  sky  is  cloudy  and  that  therefore  there 
will  be  a  storm.    Here  we  have  two  explicit  assertions, 
together  with  the  inference  involved  in  the  word  *  there- 
fore.'   It  is  of  course  a  subtle  question  for  introspection 
as  to  whether  this  threefold  assertion  really  takes  place. 
This  difficulty  does  not  at  all  affect  our  definition  of 
inference;  it  would  only  affect  the  question  whether  in 
any  given  case  inference  had  actually  occurred.    It  has 
been  suggested  that,  where  there  has  been  nothing  that 
logic  could  recognise  as  an  inference,  there  has  yet 
been  inference  in  a  psychological  sense;  but  this  con- 
tention is  absurd,  since  it  is  entirely  upon  psychological 
grounds  that  we  have  denied  the  existence  of  inference 
in  such  cases. 

Let  us  consider  further  the  logical  aspects  of  a 
genuine  inference,  following  upon  such  a  process  of 
association  as  we  have  illustrated.  The  scientist  may 
hold  that  the  appearance  of  the  sky  is  not  such  as  to 
warrant  the  expectation  of  an  on-coming  storm.    He 


INFERENCE  IN  GENERAL  5 

may,  therefore,  criticise  the  inference  as  invalid.  Thus, 
assuming  the  actuality  of  the  inference  from  the  psycho- 
logical point  of  view,  it  may  yet  be  criticised  as  invalid 
from  the  logical  point  of  view.  So  far  we  have  taken 
the  simplest  case,  where  the  single  premiss  *The  sky 
is  cloudy'  is  asserted.  But,  when  an  additional  premiss 
such  as  *In  the  past  cloudy  skies  have  been  followed 
by  storm'  is  asserted,  then  the  inference  is  further 
rationalised,  since  the  two  premisses  taken  together 
constitute  a  more  complete  ground  for  the  conclusion 
than  the  single  premiss.  This  additional  premiss  is 
technically  known  as  2.  particular  proposition.  If  the 
thinker  is  pressed  to  find  still  stronger  logical  warrant 
for  his  conclusion,  he  may  assert  that  in  all  his  expe- 
riences cloudy  skies  have  been  followed  by  storm  (a 
limited  universal).  The  final  stage  of  rationalisation  is 
reached  when  the  universal  limited  to  all  remembered 
cases  is  used  as  the  ground  for  asserting  the  unlimited 
universal  for  all  cases.  But  even  now  the  critic  may 
press  for  further  justification.  To  pursue  this  topic 
would  obviously  require  a  complete  treatment  of  induc- 
tion, syllogism,  etc.,  from  the  logical  point  of  view. 
Enough  has  been  said  to  show  that,  however  inade- 
quate may  be  the  grounds  offered  in  justification  of  a 
conclusion,  this  has  no  bearing  upon  the  nature  or  upon 
the  fact  of  inference  as  such,  but  only  upon  the  criticism 
of  it  as  valid  or  invalid. 

As  in  association,  so  also  in  perception,  a  psycho- 
logical problem  presents  itself  There  appear  to  be  at 
least  three  questions  in  dispute  regarding  the  nature  of 
perception,  which  have  close  connection  with  logical 
analysis:  First,  how  much  is  contained  in  the  percept 


i 


\»- 


6  CHAPTER  I 

besides  the  immediate  sense  experience?  Secondly, 
does  perception  involve  assertion?  Thirdly,  does  it 
involve  inference?  To  illustrate  the  nature  of  the  first 
problem,  let  us  consider  what  is  meant  by  the  visual 
perception  of  a  match-box.  This  is  generally  supposed 
to  include  the  representation  of  its  tactual  qualities ;  in 
which  case,  the  content  of  the  percept  includes  qualities 
other  than  those  sensationally  experienced.  On  the 
other  hand,  supposing  that  an  object  touched  in  the 
dark  is  recognised  as  a  match-box,  through  the  special 
character  of  the  tactual  sensations,  would  the  represen- 
tation of  such  visual  qualities  as  distinguish  a  match-box 
from  other  objects  be  included  in  the  tactual  perception 
of  it  as  a  match-box  ?  The  same  problem  arises  when 
.^  we  recognise  a  rumbling  noise  as  indicating  a  cart  in 
V-^ ^V  the  road:  i.e.  should  we  say,  in  this  case,  that  the 
auditory  percept  of  the  cart  includes  visual  or  other  dis- 
tinguishing characteristics  of  the  cart  not  sensationally 
experienced?  In  my  view  it  is  inconsistent  to  include  in 
the  content  of  the  visual  percept  tactual  qualities  not 
sensationally  experienced,  unless  we  also  include  in  the 
content  of  a  tactual  or  auditory  percept  visual  or  similar 
qualities  not  sensationally  experienced \ 

This  leads  up  to  our  second  question,  namely  whether 
in  such  perceptions  there  is  an  assertion  (a)  predicating 
of  the  experienced  sensation  certain  specific  qualities; 
or  an  assertion  {b)  of  having  experienced  in  the  past 
similar  sensations  simultaneously  with  the  perception  of 

^  In  speaking  here  of  the  menial  representation  of  qualities  not 
sensationally  experienced,  I  am  putting  entirely  aside  the  very  im- 
portant psychological  question  as  to  whether  such  mental  repre- 
sentations are  in  the  form  of  *  sense-imagery '  or  of  *  ideas.' 


^%r 


INFERENCE  IN  GENERAL  7 

a  certain  object.    Employing  our  previous  illustration, 
we  may  first  question  whether  the  assertion  '  There  is 
a  cart  in  the  road'  following  upon  a  particular  auditory 
sensation,    involves    {a)    the   explicit   characterisation 
of  that  sensation.    Now  if  the  specific  character  of  the 
noise  as  a  sensation  merely  caused  2.  visual  image  which 
in  its  turn  caused  the  assertion  'There  is  a  cart  in  the 
road/  then  in  the  absence  of  assertion  {a)  there  is  no 
explicit  inference.    In  order  to  become  inference,  the 
character  operating  (through  association)  as  cause  would 
have  to  be  predicated  (in  a  connective  judgment)  as 
ground.    On  the  other  hand,  any  experience  that  could 
be  described  as  hearing  a  noise  of  a  certain  more  or  less 
determinate  character  would  involve,  in  my  opinion, 
besides  assimilation,  a  judgment  or  assertion  \a)  expres- 
sible in  some  such  words  as  *  There  is  a  rumbling  noise.' 
The  further  assertion  that  there  is  a  cart  in  the  road 
is  accounted  for  (through  association)  by  previous  ex- 
periences of  hearing  such  a  noise  simultaneously  with 
seeing  a  cart.    Assuming  that  association  operates  by 
arousing  memory-images  of  these  previous  experiences, 
it  is  only  when  by  their  vividness  or  obtrusiveness  these 
memory-images  give  rise  to  a  memory-judgment,  that 
the  assertion  {b)  occurs.   We  are  now  in  a  position  to 
answer  the  third  question  as  to  the  nature  of  perception ; 
for,  if  either  the  assertion  of  [a)  alone  or  of  \b)  with  (a) 
occurs  along  with  the  assertion  that  there  is  a  cart  in 
the  road,  then  inference  is  involved;  otherwise  it  is  not. 
§  3.    Passing  from  the  psychological  to  the  strictly 
logical  problem,  we  have  to  consider  in  further  detail 
the  conditions  for  the  validity  of  an  inference  symbolised 
as  */  .-.  q'    These  conditions  are  twofold,  and  may  be 


8 


CHAPTER  I 


conveniently  distinguished  in  accordance  with  my  termi- 
nology as  constitutive  and  epistemic.  They  may  be 
briefly  formulated  as  follows: 

Conditions  for  Validity  of  the  Inference  'p  .-.  q' 

Constitutive  Conditions',  (i)  the  proposition  '/'  and 
(ii)  the  proposition  'p  would  imply  q'  must  both  be  true. 

Epistemic  Conditions',  (i)  the  asserting  of  'p'  and 
(ii)  the  asserting  of  'p  would  imply  q'  must  both  be 
permissible  without  reference  to  the  asserting  of  q. 

It  will  be  noted  that  the  constitutive  condition  ex- 
hibits the  dependence  of  inferential  validity  upon  a 
certain  relation  between  the  contents  of  premiss  and  of 
conclusion;  the  epistemic  condition,  upon  a  certain 
relation  between  the  asserting  of  the  premiss  and  the 
asserting  of  the  conclusion.  Taking  the  constitutive 
condition  first,  we  observe  that  the  distinction  between 
inference  and  implication  is  sometimes  expressed  by 
calling  implication  'hypothetical  inference'— the  mean- 
ing of  which  is  that,  in  the  act  of  inference,  the  premiss 
must  be  categorically  asserted ;  while,  in  the  relation  of 
implication,  this  premiss  is  put  forward  merely  hypo- 
thetically.  This  was  anticipated  above  by  rendering 
the  relation  of  implication  in  the  subjunctive  mood 
{p  would  imply  q)  and  the  relation  of  inference  in  the 
indicative  mood  {p  implies  q\ 

Further  to  bring  out  the  connection  between  the 
epistemic  and  the  constitutive  conditions,  it  must  be 
pointed  out  that  an  odd  confusion  attaches  to  the  use 
of  the  word  *  imply'  in  these  problems.  The  almost 
universal  application  of  the  relation  of  implication  in 
logic  is  as  a  relation  between  two  propositions;  but,  in 
familiar  language,  the  term  *  imply'  is  used  as  a  relation 


INFERENCE  IN  GENERAL  9 

between  two  assertions.  Consider  for  instance  {a)  ^B's 
(asserting  that  there  will  be  a  thunderstorm  would  imply 
his  having  noticed  the  closeness  of  the  atmosphere,'  and 
■(d)  *the  closeness  of  the  atmosphere  would  imply  that 
there  will  be  a  thunderstorm.'  The  first  of  these  relates 
two  mental  acts  of  the  general  nature  of  assertion,  and 
is  an  instance  of  'the  asserting  of  q  would  imply  having 
asserted/';  the  second  is  a  relation  between  two  pro- 
positions, and  is  an  instance  of  'the  proposition/  would 
imply  the  proposition  q.'  Comparing  (a)  with  (d)  we 
find  that  implicans  and  implicate  have  changed  places. 
Indeed  the  sole  reason  why  the  asserting  of  the  thunder- 
storm was  supposed  to  imply  having  asserted  the  close- 
ness of  the  atmosphere  was  that,  in  the  speaker's  judg- 
ment, the  closeness  of  the  atmosphere  would  imply  that 
there  will  be  a  thunderstorm. 

Recognising,  then,  this  double  and  sometimes  am- 
biguous use  of  the  word  'imply,'  we  may  restate  the 
first  of  the  two  epistemic  conditions  and  the  second  of 
the  two  constitutive  conditions  for  the  validity  of  the 
inference  '/  .  *.  ^'  as  follows: 

Epistemic  condition  (i) :  the  asserting  of  the  propo- 
sition '/'  should  not  have  implied  the  asserting  of  the 
proposition  '^.' 

Constitutive  condition  (ii) :  the  proposition  */'  should 
imply  the  proposition  'q! 

The  former  is  merely  a  condensed  equivalent  of  our 
original  formulation,  viz.  that  'the  asserting  of  the  pro- 
position '/'  must  be  permissible  without  reference  to  the 
asserting  of  the  proposition  'q! 

Now  the  fact  that  there  is  this  double  use  of  the 
term   'imply'  accounts  for  the  paradox  long  felt  as 


10 


CHAPTER  I 


INFERENCE  IN  GENERAL 


II 


!ti 


regards  the  nature  of  inference:  for  it  is  urged  that,  in 
order  that  an  inference  may  be  formally  valid,  it  is 
required  that  the  conclusion  should  be  contained  in  the 
premiss  or  premisses;  while,  on  the  other  hand,  if  there 
is  any  genuine  advance  in  thought,  the  conclusion  must 
not  be  contained  in  the  premiss.  This  word  'contained' 
is  doubly  ambiguous:  for,  in  order  to  secure  formal 
validity,  the  premisses  regarded  as  propositions  must 
imply  the  conclusion  regarded  as  a  proposition ;  but,  in 
order  that  there  shall  be  some  real  advance  and  not  a 
mere  petitio  principii,  it  is  required  that  the  asserting 
of  the  premisses  should  not  have  implied  the  previous 
asserting  of  the  conclusion.  These  two  horns  of  the 
dilemma  are  exactly  expressed  in  the  constitutive  and 
epistemic  conditions  above  formulated. 

§  4.  We  shall  now  explain  how  the  constitutive 
conditions  for  the  validity  of  inference,  which  have  been 
expressed  in  their  most  general  form,  are  realised  in 
familiar  cases.  The  general  constitutive  condition  */ 
would  imply  q'  informally  satisfied  when  some  specific 
logical  relation  holds  of  /  to  q\  and  it  is  upon  such  a 
relation  that  the  formal  truth  of  the  assertion  that  'p 
would  imply  q  is  based.  There  are  two  fundamental 
relations  which  will  render  the  inference  from  p  to  q, 
not  only  valid,  but  formally  valid ;  and  these  relations 
will  be  expressed  in  formulae  exhibiting  what  will  be 
called  the  Applicative  and  the  Implicative  Principles 
of  Inference.  The  former  may  be  said  to  formulate  what 
is  involved  in  the  intelligent  use  of  the  word  'every'; 
the  latter  what  is  involved  in  the  intelligent  use  of 
the  word  'if.' 

In  formulating  the  Applicative  principle,  we  take  p 


Jo  stand  for  a  proposition  universal  in  form,  and  q  for 
a  singular  proposition  which  predicates  of  some  single 
case  what  is  predicated  universally  in  /.  The  Appli- 
cative principle  will  then  be  formulated  as  follows: 

From  a  predication  about  'every'  we  may  formally 
infer  the  same  predication  about  *any  given.' 

In  formulating  the  Implicative  principle,  we  take/ 
to  stand  for  a  compound  proposition  of  the  form  'x  and 
''x  implies  j^'"  and  q  to  stand  for  the  simple  proposition 
y'   The  Implicative  principle  will  then  be  formulated 

as  follows: 

From  the  compound  proposition  'x  and  ''x  implies 
1^'"  we  may  formally  infer  'y! 

\  5.  We  find  two  different  forms  of  proposition,  one 
or  other  of  which  is  used  as  a  premiss  in  every  formal 
inference;  the  distinction  between  which  is  funda- 
mental, but  has  been  a  matter  of  much  controversy 
among  logicians.  In  familiar  logic  the  two  kinds  of 
proposition  to  which  I  shall  refer  are  known  respec- 
tively as  universal  and  hypothetical.  As  an  example  of 
the  former,  take  'Every  proposition  can  be  subjected 
to  logical  criticism';  from  this  universal  proposition  we 
may  directly  infer  'That  ''matter  exists''  can  be  sub- 
jected to  logical  criticism.'  This  inference  illustrates 
what  I  have  called  the  Applicative  Principle,  and  its 
premiss  will  be  called  an  Applicational  universal.  Take 
next  the  example  'If  this  can  swim  it  breathes,*  and  *it 
can  swim';  from  this  conjunction  of  propositions  we 
infer  that  'it  breathes';  here,  the  hypothetical  premiss 
being  in  our  terminology  called  implicative,  the  in- 
ference in  question  illustrates  the  use  of  the  Implica- 


12 


CHAPTER  I 


tive  Principle.    It  is  the  combination  of  these  two  prin- 
ciples  that  marks  the  advance  made  in  passing  from 
the  most  elementary  forms  of  inference  to  the  syllogism 
For  example:  From  ^Everything  breathes  if  able  to 
swim'  we  can  infer  ^This  breathes  if  able  to  swim' 
where  the  applicative  principle  only  is  employed.    Con- 
jommg  the  conclusion  thus  obtained  with  the  further 
premiss  'This  can  swim,'  we  can  infer  ^this  breathes  ' 
where  the  implicative  principle  only  is  employed.    In 
this  analysis  of  the  syllogism  which  involves  the  inter- 
polation  of  an  additional  proposition,  we  have  shown 
how  the  two  principles  of  inference  are  successively 
employed.    The  ordinary  formulation  of  the  syllogism 
would   read  as  follows:    ^Everything   that  can  swim 
breathes;  this  can  swim;  therefore  this  breathes.'    In 
place  of  the  usual  expression  of  the  major  premiss    I 
have  substituted  'Everything  breathes  if  able  to  swim/ 
in  order  to  show  how  the  major  premiss  prepares  the 
way  for  the  inferential  employment  successively  of  the 
applicative  and  of  the  implicative  principles. 

§  6.  Now  the  two  propositions  '  Every  proposition 
can  be  subjected  to  logical  criticism'  and  'everything 
that  IS  able  to  swim  breathes'  must  be  carefully  con 
trasted.  Both  of  them  are  universal  in  form;  but  in  the 
latter  the  subject  term  contains  an  explicit  characterising 
adjective,  viz.  able  to  swim.  The  presence  of  a  charac- 
terising adjective  in  the  subject  anticipates  the  occasion 
on  which  the  question  would  arise  whether  this  adjec- 
tive IS  to  be  predicated  of  a  given  object.  In  the 
syllogism,  completed  as  in  the  preceding  section,  the 
universal  major  premiss  is  combined  with  an  affirmative 
minor  premiss,  where  the  adjective  entertained  cate- 


INFERENCE  IN  GENERAL 


13 


rically  2.^  predicate  of  the  minor  is  the  same  as  that 
which  was  entertained  hypothetically  as  subject  of  the 
major.  This  double  functioning  of  an  adjective  is  the 
one  fundamental  characteristic  of  all  syllogism ;  where 
it  will  be  found  that  one  (or,  in  the  fourth  figure,  every) 
term  occurs  once  in  the  subject  of  a  proposition,  where 
it  is  entertained  hypothetically,  and  again  in  the  pre- 
dicate of  another  proposition  where  it  is  entertained 
categorically. 

The  essential  distinction  between  the  two  contrasted 
universals  (applicational  and  implicational)  lies  in  the 
fact  that  an  inference  can  be  drawn  from  the  former  on 
the  applicative  principle  alone,  which  dispenses  with 
the  minor  premiss.  We  have  to  note  the  nature  of  the 
substantive  that  occurs  in  the  applicational  universal  as 
distinguished  from  that  which  occurs  in  the  implicational 
universal.  The  example  already  given  contained  'pro- 
position '  as  the  subject  term,  and  a  few  other  examples 
are  necessary  to  establish  the  distinction  in  question. 
*  Every  individual  is  self-identical,'  therefore  'the  author 
of  the  Republic  is  self-identical';  'Every  conjunction 
of  predications  is  commutative,'  therefore  'the  conjunc- 
tion lightning  before  and  thunder  after  is  commutative' ; 
'  Every  adjective  is  a  relatively  determinate  specifica- 
tion of  a  relatively  indeterminate  adjective,'  therefore 
'red  is  a  relatively  determinate  specification  of  a  rela- 
tively indeterminate  adjective.'  These  illustrations 
could  be  endlessly  multiplied,  in  which  we  directly 
apply  a  universal  proposition  to  a  certain  given  instance. 
In  such  cases  the  implicative  as  well  as  the  applicative 
principle  would  have  been  involved  if  it  had  been 
necessary  or  possible  to  interpolate,  as  an  additional 


r 


^  CHAPTER  I 

datum,  a  categorical  proposition  requiring  certification 
to  serve  as  minor  premiss.    Let  us  turn  to  our  original, 
Illustration  and  examine  what  would  have  been  involved ' 
If  we  had  treated  the  inference  as  a  syllogism ;  it  would  I 
have  read  as  follows:  'Every  proposition  can  be  sub- 
jected to  logical  criticism';  That  ma^^er  exists  is  a 
proposition  ;  therefore  'That  maimer  exists  can  be  sub- 
jected to  logical  criticism.'    In  this  form,  the  substantive 
v^oxA  proposuion  occurs  as  subject   in   the  universal 
premiss,  and  as  predicate  in  the  singular  premiss.   What 
1  have  to  maintain  is  that  this  introduction  of  a  minor 
premiss  is  superfluous  and  even  misleading.    It  should 
be  observed  that,  in  all  the  illustrations  given  above  of 
the  pure  y  applicative  principle,  the  subject-term  in  the 
universal  premiss  denotes  a  general  category.   It  follows 
from  this  that  the  proposed  statement  'That  matter 
exzsts  IS  a  proposition-  is  redundant  as  a  premiss;  for  it 
IS  impossible  for  us  to  understand  the  meaning  of  the 
phrase  'matter  exists'  except  so  far  as  we  understand 
It  to  denote  a  proposition.    In  the  same  way,  it  would 
be  impossible  to  understand  the  word  'red'  without 
understanding  it  to  denote  an  adjective;  and  so  in  all 
other  cases  of  the  pure  employment  of  the  applicative 
pnnciple.    In  all  these  cases,  the  minor  premiss  which 
tTl      f'  7.^*;;"«^d  is  not  a  genuine  proposition-the 
truth   of  which   could   come   up   for   consideration- 
because  the  understanding  of  the  subject-term  of  the 
minor  demands  a  reference  of  it  to  the  general  category 
here      educated  of  it.   This  proposed  minor  premiss' 
therefore,  IS  a  peculiar  kind  of  proposition  which  is  no 
exact  y     hat  Mill  calls  'verbal,'  but  rather  what  Kan 
meant  by  'analytic,'  and  which  I  propose  to  call  'struc 


INFERENCE  IN  GENERAL 


15 


turat.''   All  structural  statements  contain  as  their  pre- 
dicate some  wide  logical  category,  and  their  fundamental 
characteristic  is  that  it  is   impossible   to  realise  the 
meaning  of  the  subject-term  without  implicitly  con- 
ceiving it  under  that  category.    The  structural  propo- 
sition can  hardly  be  called  verbal,  because  it  does  not 
4epend  upon  any  arbitrary  assignment  of  meaning  to 
word; — this  point  being  best  illustrated  by  giving 
xamples.    For  instance,  taking  as  subject-term  'the 
uthor   of   the   Republic,''   then    'The   author   of  the 
'epublic  wrote   something,'   would   be   verbal,   while 
The  author  of  the  Republic  is  an  individual,'  would 
le  structural.    In  reality  the  subject  of  a  verbal  pro- 
losition,  and  the  subject  of  a  structural  proposition  are 
not  the  same;  the  one  has  for  its  subject  t\i& phrase  'the 
author  of  the  Republic,'  and  the  other  the  object  denoted 
by  the  phrase.    This  is  the  true  and  final  principle  for 
distinguishing  a  structural  (as  well  as  a  genuinely  real 
or  synthetic  statement)  from  a  verbal  statement. 

§  7.  Since  a  category  is  expressed  always  by  a 
general  substantive  name,  the  important  question  arises 
as  to  whether  or  how  the  name  of  a  category  such  as 
•existent'  or  'proposition'  is  to  be  defined.  Now  the 
ordinary  general  substantive  name  is  defined  in  terms 
of  determinate  adjectives  which  constitute  its  connota- 
tion ;  but,  so  far  as  a  category  can  be  defined,  it  must 
be  in  terms  of  adjectival  determinables ;  e.g.  an  existent 
is  what  occupies  some  region  of  space  or  period  of 
time :  the  determinates  corresponding  to  which  would 
be,  occupying  some  specific  region  of  space  or  period  of 
time.  Similarly,  the  category  'proposition'  could  be 
defined  by  the  adjectival  determinable  'that  to  which 


i6 


CHAPTER  I 


INFERENCE  IN  GENERAL 


17 


some  assertive  attitude  can  be  adopted/  under  which 
the  relative  determinates  would  be  affirmedy  denied^ 
doubted,  etc.  We  may  indicate  the  nature  of  a  given 
category  by  assigning  the  determinables  involved  in  its 
construction.  Using  capital  letters  for  determinables 
and  corresponding  small  letters  for  their  determinates 
(distinguished  amongst  themselves  by  dashes),  the  major 
premiss  of  the  syllogism  would  assume  the  following 
form :  Every  MP  is  /  if  ;;^ ;  where  the  determinables 
M  and  P  serve  to  define  the  category  so  far  as  required 
for  the  syllogism  in  question.  Here  we  substitute  for 
the  vague  word  'thing'  previously  employed,  the  symbol 
MP  to  indicate  the  category  of  reference ;  namely,  that 
comprising  substantives  of  which  some  determinate 
character  under  the  determinables  J/ and  P  can  be  pre- 
dicated. The  statement  that  the  given  thing  is  MP  is 
redundant  where  M  and  P  are  determinables  to  which 
the  given  thing  belongs ;  for  the  thing  could  not  be  given 
either  immediately  or  in  an  act  of  construction  except 
so  far  as  it  was  given  under  the  category  defined  by  these 
determinables.  Hence  any  genuine  act  of  characterisa- 
tion of  the  thing  so  given  would  consist  in  giving  to 
these  mere  determinables  a  comparatively  determinate 
value.  For  example,  it  being  assumed  that  the  given 
thing  is  MP,  we  may  characterise  it  in  such  determi- 
nate forms  as  'm  and  p,'  'm  or  p,'  'p  if  m,'  'not  both/ 
and  ml  where  the  predication  of  the  relative  determi- 
nates m  and  /  would  presuppose  that  the  object  had 
been  constructed  under  MP.  In  defining  the  function 
of  a  proposition  to  be  to  characterise  relatively  deter- 
minately  what  is  given  to  be  characterised,  we  now  see 
that  what  is  'given    is  not  given  in  a  merely  abstract 


.sense,  but — in  being  given — the  determinables  which 
|have  to  be  determined  are  already  presupposed. 

§  8.  We  may  now  show  more  clearly  why  the  force 
of  the  term  'every'  is  distinct  from  that  of  the  term 
'if;  and  how,  in  the  syllogism,  the  two  corresponding 
principles  of  inference  are  both  involved.  The  major 
premiss  having  been  formulated  in  terms  of  the  deter- 
minables M  and  P,  the  whole  argument  will  assume 
the  following  form  : 

[a)    Every  MP  is/  if  m, 
jfrom  which  we  infer,  by  the  applicative  principle  alone: 

{b)    The  given  J//*  is/ if  ;;/. 
Next  we  introduce  the  minor,  viz. 

\c)    The  given  J//^  is  w, 
|and  finally  infer,  by  the  implicative  principle  alone: 

{d)   The  given  MP  is  /. 

jNow  if  we  held  that  the  inference  from  {a)  to  {b)  re- 

[uired  the  implicative  principle  as  well  as  the  applica- 

Itive,  so  that  a  minor  premiss  'The  given  thing  is  MP' 

lust  be  interpolated,  the  syllogism  would  assume  the 

[following  more  complicated  form : 

{a)    Everything  \^  p  \i  m  if  MP  (the  reformulated 
Imajor). 

.-.  {b)    The  given  thing  is  /   if  w  if  MP  (by   the 
[applicative  principle  alone). 

Next  we  introduce  as  minor 

{c)    The  given  thing  is  MP. 

V\{d)    The  given  thing  is  /  if  w  (by  the  implicative 
principle  alone); 

[finally,  introducing  the  original  minor,  viz. 
{e)    The  given  thing  is  m. 

;.  (/)   The  given  thing  is  /  (by  the  implicative  prin- 
[ciple  alone). 

J.  L.  II 


I  ■ 


i8  CHAPTER  I 

Now  this  lengthened  analysis  of  the  syllogism,  while 
involving  the  implicative  principle  twice,  involves  as 
well  as  the  applicative  principle  the  introduction  of  a 
new  minor,  viz.  that  the  given  thing  is  MP,  which  hints 
at  the  doubt  whether  what  is  given  is  given  as  MP. 
But  if  this  were  a  reasonable  matter  of  doubt  requiring 
explicit  affirmation,  on  the  same  principle  we  might 
doubt  whether  what  is  given  is  a  *  thing,'  in  some  more 
generic  sense  of  the  word  *thing.'    If  this  doubt  be  ad- 
mitted, the  syllogism  is  resolved  into  ^Aree  uses  of  the 
implicative  principle,  with  Iwo  extra  minor  premisses. 
Such  a  resolution  would  in  fact  lead  by  an  infinite  regress 
to  an  infinite  number  of  employments  of  the  implicative 
principle.    To  avoid  the  infinite  regress  we  must  es- 
tablish  some  principle  for  determining  the  point  at 
which  an  additional  minor  is  not  required.    The  view 
then  that  I  hold  is  not  merely  that  what  is  given  is  a 
'thing'  in  the  widest  sense  of  the  term  thing,  but  that 
what  is  given   is  always  given  as  demanding  to  be 
characterised  in  certain  definite  respects— e.g.  colour, 
size,  weight;  or  cognition,  feeling,  conation — and  that 
therefore  such  a  proposition  as   *  The  given  thing  is 
MP'  is  presupposed  in  its  being  given,  i.e.  in  being 
given,  it  is  given  as  requiring  determination  with  respect 
to  these  definite  determinates  M  and  P.    The  above 
formulation,  therefore,  in  which  the  syllogism  is  resolved 
into  a  process  involving  the  applicative  and  the  impli- 
cative principles  e3ich  only  once,  is  logically  justified; 
for  it  brings  out  the  distinction  between  the  function  of 
the  term  every  as  leading  to  the  employment  of  the 
applicative  principle  alone,  and  the  function  of  if  as 
leading  to  the  employment  of  the  implicative  principle 


INFERENCE  IN  GENERAL 


19 


lone;  and  furthermore  it  distinguishes  between  the 
|)rocess  in  inference  which  requires  the  applicative  prin- 
ciple alone  from  that  which  requires  the  implicative  as 
well  as  the  applicative  principle. 

The  distinction  between  the  cases  in  which  the  im- 
plicative principle  can  or  cannot  be  dispensed  with 

epends,  so  far,  upon  whether  the  subject-term  of  the 

niversal  stands  for  a  logical  category  or  not.  But  we 
may  go  further  and  say  that,  even  if  the  subject  of  the 

niversal  is  not  a  logical  category,  provided  that  it  is 
definable  by  certain  determinates,  and  that  the  subject 
of  the  conclusion  is  only  apprehensible  under  those 
determinables,   then  again  the  use  of  the  implicative 
principle  may  be  dispensed  with.    For  example:  'All 
material  bodies  attract;  therefore,  the  earth  attracts.' 
Here  the  term  'material  body'  is  of  the  nature  of  a 
category  in  that  it  can  only  be  defined  under  such  de- 
terminables as  'continuing  to  exist'  and  'occupying  some 
region  of  space' ;  furthermore  the  earth  is  constructively 
given  under  these  determinables:   hence  a  proposed 
minor  premiss  to  the  effect  that  the  earth  is  a  material 
body  is  superfluous,  and  the  above  inference  involves 
only  the  applicative  principle.    Again  'All  volitional  acts 
are  causally  determined;  therefore,  Socrates'  drinking 
of  hemlock  was  causally  determined.'    Here  the  subject 
of  the  conclusion  is  constructively  given  under  the  de- 
terminables involved  in  the  definition  of  volitional  act, 
which  again  justifies  the  use  of  the  applicative  principle 
alone.    As  a  third  example :  '  Every  denumerable  aggre- 
gate is  less  than  some  other  aggregate:  therefore,  an 
aggregate  whose  number  is  Xq  is  numerically  less  than 
some  other  aggregate.'    Here  the  construction  of  the 


2 — 2 


20 


CHAPTER  I 


I; 

it 


notion  of  a  class  whose  number  is  Xo  involves  its  being 
denumerable,  so  that  the  given  inference  again  re- 
quires only  the  immediate  employment  of  the  applica- 
tive principle. 

§  9.  Incidentally  the  above  analysis  of  the  major 
premiss— Every  MP  is  /  if  w— (or  still  more  simply, 
'Every  M  is  m,'  which  may  sometimes  be  true;  or 
again,  of  the  minor  premiss— 'The  given  MP  is  m  or 
'The  given  MP  is  /')— accounts  for  the  insistence  by 
certain  philosophers,  notably  Mr  Bradley,  that  every 
proposition  employs  the  relation  of  identity;  i.e.  that 
the  adjective  involved  in  the  subject  is  the  same  as  that 
involved  in  the  predicate.  This  philosophical  sugges- 
tion is,  I  hold,  true,  in  the  sense  that  the  adjectival 
determinable  in  the  subject  is  the  same  as  that  in  the 
predicate;  but  the  latter  is  a  further  determination  of 
the  former.  Now,  in  this  admission  that  the  relation  of 
identity  of  subject  to  predicate  is  involved  in  the  general 
categorical  proposition,  I  am  not  in  any  way  with- 
drawing what  was  maintained  as  regards  identity  in  my 
analysis  of  the  proposition.  For  the  identity  which  I 
denied  was  (as  it  has  been  expressed)  identity  in  deno- 
tation with  diversity  of  connotation,  i.e.  substantival 
identity  with  adjectival  diversity.  The  identity  I  have 
accepted  above  is  identity  of  an  adjectival  factor  in  the 
subject  with  an  adjectival i2.zX.0x  in  the  predicate.  More- 
over I  should  still  deny  that  the  proposition  asserts  this 
identity,  and  maintain  that  it  simply  presupposes  it,  in 
just  the  same  way  as  a  proposition  presupposes  the 
understanding  of  the  meaning  of  the  terms  involved 
without  asserting  such  meaning. 

S  10.    We  have  discussed  the  case  in  which  a  minor 


'-irt^. 


INFERENCE  IN  GENERAL 


21 


remiss  may  be  dispensed  with,  namely  that  in  which 
|a  certain  mode  of  using  the  applicative  principle  is 
tsufficient  without  the  employment  of  the  implicative, 
IWe  will  now  turn  to  a  complementary  discussion  of  the 
lease  in  which  there  is  unnecessary  employment  of  the 
mipplicative  principle,  entailed  by  the  insertion  of  what 
Jmay  be  called  a  redundant  7najor  premiss.  It  will  be 
convenient  to  call  the  redundant  minor  premiss  a  sub- 
minor,  and  the  redundant  major  premiss— to  which  we 
shall  now  turn— a  super-major.  In  this  connection  I 
shall  introduce  the  notion  of  a  formal  principle  of  in- 
ference, which  will  apply,  not  only  to  inferences  that  are 
strictly  formal,  but  also  to  inferences  of  an  inductive 
nature,  for  which  the  principle  has  not  at  present  been 
finally  formulated  and  must  therefore  be  here  expressed 
without  qualifying  detail.  The  discussion  will  deal  with 
cases  in  which  the  relation  of  premiss  or  premisses  to 
conclusion  is  such  that  the  inference  exhibits  a  formal 

principle. 

We  shall  illustrate  the  point  first  by  taking  the 
principle  of  syllogism,  and  next,  the  ultimate  (but  as  yet 
unformulated)  principle  of  induction.  As  regards  the 
syllogism,  taking  /  and  q  to  represent  the  premisses 
and  r  the  conclusion,  we  may  say  that  the  syllogistic 
principle  asserts  that  provided  a  certain  relation  holds 
between  the  three  propositions  /,  q,  and  r,  inference 
from  the  premisses  /  and  q  alone  will  formally  justify 
the  conclusion  r.  Now  it  might  be  supposed  that  this 
syllogistic  principle  constitutes  in  a  sense  an  additional 
premiss  which,  when  joined  with  /  and  q,  will  yield  a 
more  complete  analysis  of  the  syllogistic  procedure. 
But  on  consideration  it  will  be  seen  that  there  is  a  sort 


22 


CHAPTER  I 


INFERENCE  IN  GENERAL 


23 


III 


•li 


of  contradiction  in  taking  this  view :  for  the  syllogistic 
principle  asserts  that  the  premisses  /  and  q  are  alone 
sufficient  for  the  formal  validity  of  the  inference,  so  that, 
if  the  principle  is  inserted  as  an  additional  premiss  co- 
ordinate with  /  and  q,  the  principle  itself  is  virtually 
contradicted.  In  illustration  we  will  formulate  the  syllo- 
gistic principle: 

*What  can  be  predicated  of  every  member  of  a  class, 
to  which  a  given  object  is  known  to  belong,  can  be  pre- 
dicated of  that  object.' 

Now,  taking  a  specific  syllogism: 

'Every  labiate  is  square-stalked, 
The  dead-nettle  is  a  labiate, 
.'.  The  dead-nettle  is  square-stalked,* 

if  we  inserted  the  above-formulated  principle  as  a  pre- 
miss, co-ordinate  with  the  two  given  premisses,  with  a 
view  to  strengthening  the  validity  of  the  conclusion, 
this  would  entail  a  contradiction ;  because  the  principle 
claims  that  the  two  premisses  are  alone  sufficient  to 
justify  the  conclusion  *The  dead-nettle  is  square-stalked.' 
Now  the  same  holds,  mutatis  mutandis,  of  any  pro- 
posed ultimate  inductive  principle.  Here  the  premisses 
are  counted — not  as  two — but  as  many,  and  summed  up 
in  the  single  proposition  *  All  examined  instances  charac- 
terised by  a  certain  adjective  are  characterised  by 
a  certain  other  adjective';  and  the  conclusion  asserted 
(with  a  higher  or  lower  degree  of  probability)  predi- 
cates of  all  what  was  predicated  in  the  premiss  of 
all  examined.  Now,  in  accordance  with  the  inductive 
principle,  the  summary  premiss  is  sufficient  for  asserting 
the  unlimited  universal  (with  a  higher  or  lower  degree 
of  probability).   To  insert  this  principle,  as  an  additional 


jremiss  co-ordinate  with  the  summary  premiss,  would, 
therefore,  virtually  involve  a  contradiction.  In  illustra- 
:ion,  we  will  roughly  formulate  the  inductive  principle: 
OT  'What  can  be  predicated  of  all  examined  members 
%{  a  class  can  be  predicated,  with  a  higher  or  lower 
degree  of  probability,  of  all  members  of  the  class.' 

:iNow,  taking  a  specific  inductive  inference: 
,  *A11  examined  swans  are  white.  .*.  With  a  higher 
[or  lower  degree  of  probability,  all  swans  are  white,' 
if  we  inserted  the  above-formulated  inductive  principle 
as  a  premiss,  co-ordinate  with  the  summary  premiss  'AH 
examined  swans  are  white,'  with  a  view  to  strengthening 
the  validity  of  the  conclusion,  this  would  entail  a  con- 
tradiction ;  because  the  principle  claims  that  this  summary 
[premiss  is  alone  sufficient  to  justify  the  conclusion  that 
'With  a  higher  or  lower  degree  of  probability,  all  swans 

I  are  white.* 

We  may  shortly  express  the  distinction  between  a 
principle  and  a  premiss  by  saying  that  we  draw  the 
conclusion  from  the  premisses  in  accordance  with  (or 
through)  the  principle.  In  other  words,  we  immediately 
see  that  the  relation  amongst  the  premisses  and  con- 
clusion is  a  specific  case  of  the  relation  expressed  in  the 
principle,  and  hence  the  function  of  the  principle  is  to 
stand  as  a  universal  to  the  specific  inference  as  an  in- 
stance of  that  universal :  where  the  latter  may  be  said 
to  be  inferred  from  the  former  (if  there  is  any  genuine 
inference)  in  accordance  with  the  Supreme  Applicative 
principle.  For  example :  from  x  =y  and  jv  =  ^,  we  may 
infer  x  =  2.  This  form  of  inference  is  expressed,  in 
general  terms,  in  the  Principle:  'Things  that  are  equal 
to  the  same  thing  are  equal  to  one  another.'   Now,  here, 


i 

! 
I 


24 


CHAPTER  I 


INFERENCE  IN  GENERAL 


25 


the  two  premisses — x^y2indy  =  2 — zx^  alone  sufficient 
for  the  conclusion  x  =  z\  the  conclusion  being  drawn 
from  the  two  premisses  through  or  in  accordance  with 
the  principle  which  states  that  the  two  premisses  are 
alone  sufficient  to  secure  validity  for  the  conclusion. 
The  principle  cannot  therefore  be  added  co-ordinately 
to  the  premisses  without  contradiction.  Moreover  the 
above-formulated  principle  (which  expresses  the  tran- 
sitive property  of  the  relation  of  equality)  cannot  be 
subsumed  under  the  syllogistic  principle.  In  the  same 
way  the  syllogistic  or  inductive  principle  may  be  called 
a  redundant  or  super-major,  because  it  introduces  a  mis- 
leading or  dispensable  employment  of  the  applicative 
principle. 

§  II.  There  is  a  special  purpose  in  taking  the  in- 
ductive and  syllogistic  principles  in  illustration  of  super- 
majors,  for  many  logicians  have  maintained  that  any 
specific  inductive  inference  does  not  rest  on  an  inde- 
pendent principle,  but  upon  the  syllogistic  principle 
itself;  in  other  words,  they  have  taken  syllogism  to 
exhibit  the  sole  form  of  valid  inference,  to  which  any 
other  inferential  processes  are  subordinate.  Now  it  is 
true  that  the  inductive  principle  could  be  put  at  the 
head  of  any  specific  inductive  inference,  and  thus  be 
related  to  the  specific  conclusion  as  the  major  premiss 
of  a  syllogism  is  related  to  its  conclusion ;  but  the  same 
could  be  said  of  the  syllogistic  principle :  namely  that  it 
could  be  put  at  the  head  of  any  specific  syllogistic  in- 
ference to  which  it  is  related  in  the  same  way  as  the 
major  premiss  of  a  syllogism  is  related  to  its  conclusion. 
But,  if  we  are  further  to  justify  the  specific  inductive 
inference  by  introducing  the  inductive  principle,  then, 


by  parity  of  reasoning,  we  should  have  to  introduce  the 
syllogistic  principle  further  to  justify  the  specific  syllo- 
gistic inference.  But  in  the  case  of  the  syllogism  this 
would  lead  to  an  infinite  regress  as  the  following  illus- 
tration will  show.  Thus,  taking  again  as  a  specific 
syllogism,  that 

from  (/>)  *A11  labiates  are  square-stalked' 
and  {q)  'The  dead-nettle  is  a  labiate' 
we  may  infer  (r)  'The  dead-nettle  is  square-stalked,' 

and,    adding    to    this    as    super-major    the    syllogistic 

principle,  namely  (a),  we  have  the  following  argument : 

[a)  Foreverycaseof  J/,  of^'andof/*:  the  inference 
'every  M  is  P,  and  5  is  J/,  .-.  6*  is  /*'  is  valid. 

{b)  The  above  specific  syllogism  is  a  case  of  {a). 

(c)  .'.  The  specific  syllogism  is  valid. 
But  here,  in  inferring  from  (a)  and  [6)  together  to  (c), 
we  are  employing  the  syllogistic  principle,  which  must 
stand  therefore  as  a  super-major  to  the  inference  from 
(a)  and  (6)  together  to  (c),  and  therefore  as  super-super- 
major  to  the  specific  inference  from/  and  ^  to  r.  This 
would  obviously  lead  to  an  infinite  regress. 

We  may  show  that  a  similar  infinite  regress  would 
be  involved  if  we  introduced,  as  super-major,  the  in- 
ductive principle,  by  the  following  illustration.  Taking 
again  as  a  specific  inductive  inference  that  from  'All 
examined  swans  are  white'  we  may  infer  with  a  higher 
or  lower  degree  of  probability  that  '  All  swans  are 
white';  and  adding  to  this  as  super-major  the  in- 
ductive principle,  namely  (a),  we  have  the  following 
argument: 

(a)  For  every  case  of  Af  and  of  P:  from  'every 
examined  M  is  P'  we  may  infer,  with  a  higher  or  lower 
degree  of  probability,  that  'every  M  is  P'; 


26 


CHAPTER  I 


{b)  The  above  specific  induction  is  a  case  of  {a), 
(c)  .-.  The  specific  induction  is  valid. 

But,  here  we  may  argue  in  regard  to  this  (a),  (6),  (c)  as 
in  the  case  of  the  previous  (a),  (6),  (c).  Thus,  by  in- 
troducing the  inductive  principle  as  a  redundant  major 
premiss,  we  shall  be  led  as  before,  by  an  infinite  regress, 
to  a  repeated  employment  of  the  syllogistic  principle. 

This  whole  discussion  forces  us  to  regard  the  in- 
ductive and  syllogistic  principles  as  independent  of  one 
another,  the  former  not  being  capable  of  subordination 
to  the  latter;  for  we  cannot  in  any  way  deduce  the  in- 
ductive principle  from  the  syllogistic  principle.  Those 
who  have  regarded  the  syllogistic  principle  as  ultimately 
supreme,  have  in  fact  arrived  at  this  conclusion  by  noting 
that,  as  shown  above,  the  inductive  principle  could  be 
introduced  as  a  major  for  any  specific  inductive  inference, 
in  which  case  the  inference  would  assume  the  syllogistic 
form  (a),  {b\  (c).  But  this  in  no  way  affects  the  supremacy 
of  the  inductive  principle  as  independent  of  the  syllo- 
gistic. 


CHAPTER  II 

'HE  RELATIONS  OF  SUB-ORDINATION  AND  CO-ORDI- 
NATION AMONGST  PROPOSITIONS  OF  DIFFERENT 
TYPES 

§  I.  In  the  previous  chapter  we  have  shown  that  the 
[syllogism  which  establishes  material  conclusions  from 
material  premisses  involves  the  alternate  use  of  the 
Applicative  and  Implicative  principles.  Now  these  two 
principles,  which  control  the  procedure  of  deduction  in 
its  widest  application,  are  required  not  only  for  material 
[inferences,  but  also  for  the  process  of  establishing  the 
formulae  that  constitute  the  body  of  logically  certified 
theorems.  All  these  formulae  are  derived  from  certain 
intuitively  evident  axioms  which  may  be  explicitly 
enumerated.  It  will  be  found  that  the  procedure  of 
deducing  further  formulae  from  these  axioms  requires 
only  the  use  of  the  Applicative  and  Implicative  prin- 
ciples ;  these,  therefore,  cover  a  wider  range  than  that 
of  mere  syllogism.  But  a  final  question  remains,  as  to 
how  the  formal  axioms  are  themselves  established  in 
their  universal  form.  By  most  formal  logicians  it  is 
assumed  that  these  axioms  are  presented  immediately 
as  self-evident  in  their  absolutely  universal  form ;  but 
such  a  process  of  intuition  as  is  thereby  assumed  is 
really  the  result  of  a  certain  development  of  the  reasoning 
powers.  Prior  to  such  development,  I  hold  that  there 
is  a  species  of  induction  involved  in  grasping  axioms  in 
their  absolute  generality  and  in  conceiving  of  form  as 


28 


CHAPTER  II 


constant  in  the  infinite  multiplicity  of  its  possible  appli- 
cations.  We  therefore  conclude  that  behind  the  axioms 
there  are  involved  certain  supreme  principles  which  bear 
to  the  Applicative  and  Implicative  principles  the  same 
relation  as  induction  in  general  bears  to  deduction;  and, 
even  more  precisely,  that  these  two  new  principles  may 
be  regarded  as  inverse  to  the  Applicative  and  Impli- 
cative principles  respectively.    This  being  so,  it  will  be 
convenient  to  denominate  them  respectively.  Counter- 
applicative  andCounter-implicative.  It  should  be  pointed 
out  that  whereas  the  Applicative  and  Implicative  prin- 
ciples hold  for  material  as  well  as  formal   inferential 
procedure,    the   Counter-principles   are    used    for   the 
establishment  of  the  primitive  axioms  themselves  upon 
which  the  formal  system  is  based.    We  will  then  pro- 
ceed to  formulate  the  Counter-principles,  each  in  imme- 
diate  connection  with  its  corresponding  direct  principle. 
§  2.  The  Applicative  principle  is  that  which  justifies 
the  procedure  of  passing  from  the  asserting  of  a  pre- 
dication about  '  every '  to  the  asserting  of  the  same 
predication  about  'any  given.'    Corresponding  to  this, 
the  Counter-applicative  principle  may  be  formulated: 

'When  we  are  justified  in  passing  from  the  asserting 
ot  a  predication  about  some  one  given  to  the  asserting 
o  the  same  predication  about  some  other,  then  we  are 
also  justified  in  asserting  the  same  predication  about 
every,  ^ 

Roughly  the  Applicative  principle  justifies  inference 
from  ^ every'  to  'any,'  and  the  Counter-applicative 
justifies  inference  from  'any'  to  'every';  but  whereas 
the  former  principle  can  be  applied  universally,  the 
latter  holds  only  in  certain  narrowly  limited  cases;  'and, 


SUB-ORDINATION  AMONGST  PROPOSITIONS       29 

n  particular,  for  the  establishment  of  the  primitive 
formulae  of  Logic.  These  cases  may  be  described  as 
:hose  in  which  we  see  the  universal  in  the  particular, 
ind  this  kind  of  inference  will  be  called  'intuitive  in- 
luction,'  because  it  is  that  species  of  generalisation  in 
^hich  we  intuite  the  truth  of  a  universal  proposition  in 
^he  very  act  of  intuiting  the  truth  of  a  single  instanced 
ince  intuitive  induction  is  of  course  not  possible  in 
ivery  case  of  generalisation,  we  have  implied  in  our 
Formulation  of  the  principle  that  the  passing  from  '  any ' 
CO  'every'  is  justified  only  when  the  passing  from  'any 
►ne'  to  'any  other'  is  justified.  Now  there  a^^e  forms  of 
inference  in  which  we  can  pass  immediately  from  any 
ine  given  case  to  any  other ;  if  it  were  not  so,  the 
principle  would  be  empty.  For  instance,  we  may  illus- 
:rate  the  Applicative  principle  by  taking  the  formula: 
For  every  value  of/  and  of  q,  "/  and  ^"  would  imply 
"/",'  from  which  we  should  infer  that  'thunder  and 
lightning'  would  imply  'thunder.'  If  now  we  enquire 
Ihow  we  are  justified  in  asserting  that  for  every  value 
of/  and  of  ^,  'p  and  q'  would  imply  '/,'  the  answer 
will  supply  an  illustration  of  the  Counter-applicative 
principle.  Thus,  in  asserting  that  '"thunder  and  light- 
ning" would  imply  "thunder"'  we  see  that  we  could 
proceed  to  assert  that  '  "blue  and  hard"  would  imply 
"blue",'  and  in  the  same  act,  that  '  "/  and  q''  would 
imply  "/"  for  all  values  of/  and  of  ^.' 

§  3.  The  second  inverse  principle  to  be  considered  is 
the  Counter-implicative.  Before  discussing  this  inverse 
principle,  it  will  be  necessary  to  examine  closely  the 

^  This  is  a  special  case  of  *  intuitive  induction,'  the  more  general 
uses  of  which  will  be  examined  in  Chapter  VIII. 


30 


CHAPTER  II 


Implicative  principle  itself,  which  may  be  provisionally 
formulated:  'Given  that  a  certain  proposition  would 
formally  imply  a  certain  other  proposition,  we  can  validly 
proceed  to  infer  the  latter  from  the  former.'  Now  we 
find  that  the  one  positive  element  in  the  notion  of 
formal  implication  is  its  equivalence  to  potentially  valid 
inference,  and  that  there  is  no  single  relation  properly 
called  the  relation  of  implication.  We  must  therefore 
bring  out  the  precise  significance  of  the  Implicative 
principle  by  the  following  reformulation:  *  There  are 
certain  specifiable  relations  such  that,  when  one  or 
other  of  these  subsists  between  two  propositions,  we 
may  validly  infer  the  one  from  the  other.'  From  the 
enunciation  of  this  principle  we  can  pass  immediately 
to  the  enunciation  of  its  inverse — the  Counter-implica- 
tive  principle: 

'When  we  have  inferred,  with  a  consciousness  of 
validity,  some  proposition  from  some  given  premiss  or 
premisses,  then  we  are  in  a  position  to  realise  the  specific 
form  of  relation  that  subsists  between  premiss  and  con- 
clusion upon  which  the  felt  validity  of  the  inference 
depends.' 

Here,  as  in  the  case  of  the  Counter-applicative  principle, 
we  must  point  out  that  there  are  cases  in  which  we  in- 
tuitively recognise  the  validity  of  inferring  some  con- 
crete conclusion  from  a  concrete  preimiss,  before  having 
recognised  the  special  type  of  relation  of  premiss  to 
conclusion  which  renders  the  specific  inference  valid ; 
otherwise  the  Counter-implicative  principle  would  be 
empty.  In  illustration,  we  will  trace  back  some  accepted 
relation  of  premiss  to  conclusion,  upon  which  the  validity 
of  inferring  the  one  from  the  other  depends ;  and  this 


SUB-ORDINATION  AMONGST  PROPOSITIONS      31 

will  entail  reference  to  a  preliminary  procedure  in  ac- 
cordance with  the  Counter-applicative  principle;  for 
every  logical  formula  is  implicitly  universal.  Thus  we 
might  infer,  with  a  sense  of  validity  from  the  information 
*Some  Mongols  are  Europeans'  and  from  this  datum 
alone,  the  conclusion  'Some  Europeans  are  Mongols.' 
We  proceed  next  in  accordance  with  the  Counter-appli- 
cative principle  to  the  generalisation  that  the  inference 
from  *  Some  M  \s  P'  X.o  '  Some  P  is  M'  is  always  valid. 
Finally  we  are  led,  in  accordance  with  the  Counter- 
implicative  principle,  to  the  conclusion  that  it  is  the  re- 
lation of  'converse  particular  affirmatives'  that  renders 
the  inference  from  'Some  J/  is  /*'  to  *Sorne  P  is  M* 
valid. 

§  4.  We  have  regarded  the  intuition  underlying  the 
Counter-applicative  principle  as  an  instance  of  'seeing 
the  universal  in  the  particular';  and  correspondingly  the 
intuition  underlying  the  Counter-implicative  principle 
may  be  regarded  as  an  instance  of  'abstracting  a  common 
form  in  diverse  matter.'  But  the  direct  types  of  intuition 
operate  over  a  much  wider  field  than  the  Counter-appli- 
cative and  Counter-implicative  principles :  for,  whereas 
the  twin  inverse  principles  operate  only  in  the  estab- 
lishment of  axioms,  the  direct  types  of  intuition 
are  involved  wherever  there  is  either  universality  or 
form.  These  direct  types  of  intuition  have  been  ex- 
plicitly recognised  by  philosophers ;  but  the  still  more 
purely  intuitive  nature  of  the  procedure  conducted  in 
accordance  with  the  twin  inverse  principles  accounts  for 
the  fact  that  these  principles  have  hitherto  not  been 
formulated  by  logicians.  Moreover  the  point  of  view 
from  which  the  inverse  principles  have  been  described 


32 


CHAPTER  II 


and  analysed  is  purely  epistemic,  and  the  epistemic 
aspect  of  logical  problems  has  generally  been  ignored 
or  explicitly  rejected  by  logicians.  It  follows  also  from 
their  epistemic  character  that  these  principles,  unlike 
the  Applicative  and  Implicative  principles  of  inference, 
cannot  be  formulated  with  the  precision  required  for  a- 
purely  mechanical  or  blind  application. 

§  5.  The  operation  of  these  four  supreme  principles 
is  best  exhibited  by  means  of  a  scheme  which  comprises 
propositions  of  every  type  in  their  relations  of  super-, 
sub-,  or  co-ordination  to  one  another.  We  propose, 
therefore,  to  devote  the  remainder  of  this  chapter  to 
the  construction  and  elucidation  of  such  a  scheme. 

I.  Superordinate  Principles  of  Inference, 

I  a.    The  Counter-applicative  and  Counter-impli- 
cative. 

\b.    The  Applicative  and  Implicative. 

II.  Formulae',  i.e.  formally  certified  propositions 
expressible  in  terms  of  variables  having  general 
application. 

\\a.    Primitive    formulae    (or    axioms)    derived 
directly  from  II I ^  in  accordance  with  \a, 

\\b.    Formulae  successively  derived  from  11^  by 
means  of  \b, 

III.  Formally  Certified  Propositions  expressed  in 
terms  having  fixed  application. 

\\\a.    Those  from  which  \\a  are  derived  by  use 
of  the  principles  \a. 

\\\b.    Those  which  are  derived  from  1 1  <5  by  use 
of  the  Applicative  principle  I  b, 

IV.  Experientially  Certified  Propositions, 
IVtf.    Data  directly  certified  in  experience. 


SUB-ORDINATION  AMONGST  PROPOSITIONS       33 

IV^.  Concrete  conclusions  inferred  from  IV^  by 
means  of  implications  of  the  type  III,  and 
therefore  established  in  accordance  with  the 
Implicative  principle,  I^. 

I.    The  highest  type  consists  of  those  principles 
under  one  or  other  of  which  every  inference  is  sub- 
ordinated.   These  superordinate  principles  consist  of 
la:  the  Counter-applicative  and  Counter-implicative, 
to  which  intuitional  inferences  are  subordinated:  and 
oi  \b\  the  Applicative  and  Implicative,  to  which  de- 
monstrative inferences  are  subordinated.     \a  are  those 
principles  in  accordance  with  which  the  primitive  for- 
mulae (or  axioms)  of  Logic  are  established.    But  the 
choice  of  logical  formulae  that  are  accounted  primitive 
is  (within  limits)  arbitrary,  and  since  any  comparati\  ( ly 
self-evident  logical  formula,  instead  of  being  exhibited 
as  derivative,  could  be  regarded  as  established  directly 
in  accordance  with  these  inverse  principles,  their  scope 
must  not  be  restricted  to  the  establishment  of  the  more 
or  less  arbitrarily  selected  axioms.    !  t  will  be  found  later, 
when  we  discuss  the  types  of  proposition  m  level  III, 
that  the  content  or  material  upon  which  the  inverse  prin- 
ciples \a  operate,  is  supplied  by  the  propositions  of  type 
Ilia.    On  the  other  hand,  the  Applicative  and  Implica- 
tive principles  \b  stand  in  the  relation  of  immediate 
superordination  to  the  processes  of  inference  by  which 
from  1 1  a  are  derived  1 1  b,  viz.  the  general  formulae  of 
deduction,  induction,  demonstration,  probability,  etc. 

1 1.  The  characteristic  common  to  all  the  propositions 

on  the  second  level  is  ttuit  ilic }/  are  formally  certified, 

and  are  expressible  in  terms  of  variable  symbols.   They 

are  theoretically  infinite  in  number,  and  may  be  divided 

J.  L.  n  3 


34 


CHAPTER  II 


into  two  groups,  *  primitive'  and  'derivative';  but,  as 
pointed  out  above,  the  line  of  demarcation  between  the 
two  cannot  be  sharply  drawn.    Thus  1 1  a  comprises  a 
small  number  of  primitive  formulae  which  are  directly 
established  in  accordance  with  the  twin  inverse  principles 
\a:  for  example,  the  commutative  and  associative  laws, 
the  laws  of  identity  and  of  negation,  the  modus  ponendo 
tollens,  etc.,  or  such  of  these  as  have  been  selected  as 
primitive.    Next,   11^  comprises  an  indefinite  number 
of  formulae  successively  derived  from  the  primitive  for- 
mulae 11^:  for  example,  the  dictum  of  the  syllogism, 
and  other  more  complicated  logical  formulae,  as  well  as 
the  rules  of  arithmetic  and  algebra.   All  the  formulae  of 
level  1 1  are  implicitly  universal  in  form ;  and  most  of 
those  that  are  logical  (as  distinct  from  mathematical) 
assert  relations  of  implication.    Each  formula  in  11^  is 
derived  from  previously  certified  formulae,  and  ultimately 
from  those  in  1 1^,  the  process  of  derivation  being  marked 
at  each  step  by  the  relation  *  therefore.'    Now  wherever 
a  previously  certified  relation  of  implication  is  used  for 
deriving  a  new  formula  (in  which  case  its  implicans 
must  also  have  been  previously  certified  in  order  that 
its  implicate  may  be  derivatively  certified)  the  procedure 
is  conducted  in  accordance  with  the  implicative  principle, 
to  which  therefore  all  such  cases  of  inference  are  to  be 
subordinated.    Again,  the  process  of  successive  deriva- 
tion of  the  formulae  of  1 1  ^  entails  explicit  recognition 
of  the  implicit  universality  of  the  formulae  from  which 
they  are  derived ;  and  this  allows  us,  by  means  of  the 
Applicative  principle,  to  replace  the  illustrative  symbols 
occurring  in  an  earlier  formula  by  any  other  symbols, 
in  order  to  derive  a  new  formula. 


\ 


\ 


SUB-ORDINATION  AMONGST  PROPOSITIONS       35 

III.    The  third  level  contains  formally  certified  pro- 
positions expressed  entirely  in  linguistic  terms  of  fixed 
application  ;  and,  like  its  predecessors,  is  to  be  divided 
into  two  sections,  the  division  being  made  on  precisely 
the  same  grounds  as  that  between  \\a  and  11^.    Thus 
the  propositions  of  Illa  constitute  the  intuited  material 
for  deriving  11^  in  accordance  with  the  inverse  prin- 
ciples \a]  and  the  propositions  of  I II <5  are  exhibited 
as  derived  from  1 1  ^  in  accordance  with  the  applicative 
principle  \b\    It  will  be  seen,  however,  that  the  relation 
of  IWaio  \Ub  differs  from  that  of  llato  Il^in  that  the 
two  parts  of  III  are  not  inferentially  connected,  as  are 
those  of  II.     The  propositions  comprised  in  III<J  are 
obtained  from    11^  by  substituting  words  with  fixed 
application  for  the   variable   symbols  ;  these  proposi- 
tions,  then,   are  specialised  instances  of  the  general 
formulae  which  constitute  the  second  level,  and  are 
established  from  them  in  accordance  with  the  appli- 
cative principle  alone.    Any  logical  text-book  teems 
with  examples  of  this  procedure,  where  instances  under 
such  formulae  as  the  modus  tollendo  tollens,    or  the 
syllogistic  dictum  are  represented  in  words  with  fixed 
application,    and    then    exhibited   as  derived   (in   ac- 
cordance with  the  applicative  principle)  from  the  appro- 
priate general  formula.    It  is  usual  in  these  cases,  how- 
ever, to  exhibit  the  conclusion  as  being  inferred  from 
the  premisses,  thus  leading  the  reader  to  suppose  that 
it  is  the  conclusion  which  has  been  formally  certified, 
whereas,  properly  speaking,  what  has  been  formally 
certified  is  the  relation  of  implication  of  premisses  to 

^  Hence  the  point  of  division  between  Ilia  and  III^  cannot  be 
precisely  indicated. 

3—2 


I 


36 


CHAPTER  II 


conclusion.  It  will  be  found  below  that  this  distinction 
between  implication  and  inference  is  the  essential  con- 
sideration in  comparing  1 11^  with  IV^. 

IV.  The  fourth  and  lowest  level  consists  of  experi- 
entially  Certified  propositions  expressed  in  concrete 
terms  ;  and  again  this  level  must  be  divided  into  two 
sections,  viz.  IV^  the  primitives  and  Wb  the  deriva- 
tives, these  two  sections  standing  in  a  relation  to  one 
another  which  in  every  respect  agrees  with  the  rela- 
tion of  11^  to  11^.  Thus  the  propositions  comprised 
in  Wb  are  successively  derived  from  experiential 
propositions  that  have  been  previously  certified,  and 
ultimately  derived  from  the  primitive  experiential  data 
which  constitutes  IV^.  And  again,  as  in  the  case  of 
formally  certified  propositions,  here,  in  that  of  ex- 
perientially  certified  propositions,  the  point  of  division 
between  the  primitives  and  derivatives  is  not  precisely 
fixed  ;  the  primitives  of  IV,  like  those  of  II,  are  sup- 
posed to  be  intuitively  accepted,  i.e.  in  this  case  per- 
ceptually guaranteed ;  but  philosophers  do  not  agree 
on  the  question  of  the  kind  and  range  of  experiences 
that  can  be  regarded  as  in  this  case  immediate*  More- 
over, as  regards  experiential  propositions  admittedly 
derivative  and  not  primitive,  no  logician  or  philosopher 
has  as  yet  been  able  to  show  how  they  can  be  exhibited 
as  derived  ultimately  from  absolutely  primitive  data  of 
experience.  Hence,  in  expounding  the  logical  nature 
of  the  propositions  in  this  lowest  level,  attention  must 
be  chiefly  directed  to  the  mode  in  which  any  admittedly 
derived  proposition  is  inferred  from  some  previously 
certified  proposition,  without  enquiring  too  closely  as 
to  the  mode  in  which  the  previous  certification  had 


SUB-ORDINATION  AMONGST  PROPOSITIONS      37 

been  conducted,  or  whether  this  certification  could 
properly  be  called  perceptually  immediate.  The  mode 
of  deriving  an  experiential  conclusion  from  experi- 
entially  certified  premisses  may  be  explained  quite 
briefly  ;  the  former  is  derived  directly  from  the  latter 
by  means  of  some  implication  of  type  III,  of  which  the 
implicans  is  composed  of  the  previously  certified  pre- 
misses and  the  implicate  is  the  conclusion  required. 
Since  in  this  process  a  relation  of  implication  is  trans- 
formed into  the  relation  *  therefore,'  it  is  obvious  that 
the  implicative  principle  alone  is  employed.  But,  to 
complete  the  exposition,  we  must  trace  the  process  of 
derivation  one  stage  further  back,  namely  to  the  general 
formulae  of  line  II.  Thus,  while  any  conclusion  in  IV^ 
is  directly  derived  from  premisses  I  V<a:  by  means  of  an 
implicative  proposition  of  the  type  III,  and  so  far 
employs  the  implicative  principle  alone  ;  yet,  since  any 
proposition  of  type  III  is  itself  derived  from  some 
formula  of  type  1 1  in  accordance  with  the  applicative 
principle  alone,  it  follows  that  both  these  principles  are 
jointly  involved  in  deriving  experiential  conclusions 
from  experiential  data.  This  mode  of  derivation  is 
illustrated  in  any  text-book  example  of  a  concrete 
syllogism,  where  from  previous  experiential  certification 
of  the  premisses  we  infer  the  experiental  certification 
of  the  conclusion.  For  the  sake  of  variety  we  will 
choose,  for  illustrating  the  processes  of  deriving  any 
conclusion  IV^,  the  formula  of  pure  induction,  which, 
as  was  maintained  in  the  preceding  chapter,  must  be 
included  amongst  the  formulae  constituting  the  second 
level.  Take  for  instance  as  premiss  :  *  Every  examined 
case  of  an  acquired  characteristic  is  non-transmitted/ 


38 


CHAPTER  II 


This  datum  is  regarded  not,  of  course,  as  a  mere  sum- 
mary of  directly  given  experiences,  but  as  the  product 
of  various  constructive  and  inferential  processes  which 
may  be  supposed  ultimately  to  be  based  on  sense-data/ 
Now  by  means  of  the  concrete  implication  that  '  every 
examined  case  of  an  acquired  characteristic  being  non- 
transmitted  would  imply,  with  a  certain  degree  of  pro- 
bability, that  no  acquired  characteristic  is  transmissible,* 
conjoined  with  the  certified  fact  that  '  in  all  examined 
cases  acquired  characteristics  are  non-transmitted,'  we 
infer  the  conclusion  that   *with  a  certain    degree  of 
probability  no  acquired  characteristic  is  transmissible/ 
In  this  fourth  line,  we  are  representing  propositions 
as  proved,  or  as  validly  asserted  on  the  basis  of  ex- 
periential knowledge,  and  this  suggests  an  ambiguity 
in  the  use  of  the  term  *  ground '  which  is  sometimes 
applied  in  philosophy  to  the  experiential  data  which 
maybe  said  to  be  co-ordinate  with  the  experiential  con- 
clusions; the  same  term  Aground'  being  also  applied  to 
the  logical  formulae  of  induction  or  deduction  which 
are  superordinate  to   the  experiential   data  and  con- 
clusions.   This  ambiguity  in  the  use  of  the  term   is 
removed  by  thus  recognising  the  distinction  between 
superordinate  and  co-ordinate. 

§  6.  In  further  elucidation  of  the  scheme,  we  will 
show  what  exactly  is  involved  in  level  II,  where  em- 
phasis has  been  put  upon  the  variable  symbols.  In 
logical  text-books  we  find  that  an  inference  or  implica- 
tion is  expressed  in  terms  of  variable  symbols,  such  as 
5,  M,  and  P,  and  this  is  always  supplemented  by  a 
formula  expressed  entirely  linguistically,  but  which  is 
its  mere   equivalent.     For   example,    it  may  be  first 


5 


I 


SUB-ORDINATION  AMONGST  PROPOSITIONS       39 

asserted  that  '  Every  P  is  Q  would  imply  that  some  Q 
is  P ' ;  and  here  the  assertion  of  implication  is  under- 
stood as  being  implicitly  universal,  i.e.  that  it  holds  for 
'all  values  of  P  and  Q,    This  is  usually  supplemented  by 
the  so-called  'Rule  for  the  Conversion  of  ^,'  viz.  that 
*  Any  universal  affirmative  proposition  would  imply  the 
particular  affirmative  obtained  by  interchanging  subject 
and  predicate  terms.'    But  this  is  merely  an  alternative 
formulation,   and    is    not  related   to  the  former  as  a 
universal  to  its  instance.    We  see  therefore  that  the 
formulae  of  level  II   are  not  necessarily  expressed  in 
terms  of  variables,  but  may  be  expressed  with  precise 
equivalence  in  linguistic  terms  only.    The  possibility  of 
this  linguistic  formulation  depends  upon  the  invention 
of  a  technical  terminology  which  employs  such  terms  as 
subject,   predicate,  conversion,   universal,  proposition, 
etc.     The  reason   why  what  is  called  symbolic  logic 
requires  the  employment  of  variable  symbols  is  essen- 
tially because  the  logical  formulae  which  it  establishes 
are  so  complicated  that  a  terminology  could  hardly  be 
invented  for  dealing  with  them.    There  is  therefore  no 
difference  of  principle  involved  in  the  employment  of 
variable  symbols  by  symbolic  logic  and  the  employ- 
ment of  technical  linguistic  terms  by  ordinary  logic. 
By  the  employment  of  the  technical  terminology  of 
logic  the  variables  entering  into  any  formula  are  elimi- 
nated en  bloc,  leaving  the  formula  with  the  same  range 
of  universality  as  before.     In  contrast  with  this,  a  pro- 
position of  level  III,  being  obtained  from  level  II  by 
replacing  each  of  the  several  variables  by  a  particular 
word  of  fixed  application,  constitutes  a  single  instance 
of  the   general  formula.     For   instance,    *that  every 


40 


CHAPTER  II 


trespasser  will  be  prosecuted  would  imply  that  some 
prosecuted  person  is  a  trespasser  \'  is  a  specific  assertion 
obtained  by  the  applicative  principle  from  the  universal 
formula  of  conversion  adduced  above. 

This  last  discussion  of  the  distinction  and  connection 
between  the  use  of  variable  symbols  and  that  of  linguistic 
terminology,  points  to  certain  respects  in  which  the 
methods  of  symbolic  logic  differ,  and  others  in  which 
they  agree  with  those  of  ordinary  logic — a  topic  which 
will  be  treated  at  greater  length  in  the  following  chapter. 

^  This  illustration  is  chosen  in  order  incidentally  to  suggest  that 
the  text-books  are  not  always  infallible,  the  form  of  implication  in 
question  being  at  least  dubious. 


'i 


CHAPTER  III 

SYMBOLISM  AND  FUNCTIONS 

§  I.  The  value  of  symbolism,  as  is  universally  re- 
cognised, is  due  to  the  extreme  precision  which  its 
employment  affords  to  the  process  of  logical  demonstra- 
tion. As  a  language  it  differs  from  all  ordinary  languages 
in  three  respects,  viz.  systematisation,  brevity  and 
exactness ;  and  in  these  respects  differs  from  all  other 
languages  in  a  way  in  which  they  do  not  differ  from  one 

another. 

Now,  when  we  examine  the  language  of  symbolism, 
we  find  that  symbols  are  of  two  fundamentally  distinct 
kinds,  which  I  propose  to  call  illustrative  and  shorthand. 
In  such  familiar  logical  forms  as  'S\sP,'  '  Every  M  is 
P,'  etc.,  S,  My  P,  exemplify  illustrative  symbols.  Thus 
an  illustrative  symbol  is  represented  by  a  single  letter 
chosen  from  some  alphabet.  Shorthand  symbols,  on 
the  other  hand,  are  mere  substitutes  for  words,  and 
serve  the  obvious  purpose  of  saving  time  in  reading 
and  space  in  writing.  Some  of  them,  in  fact,  are  literal 
abbreviations,  such  as  *rel.'  for  'relative,'  'prop.*  for 
'proposition,'  'indiv.' for 'individual.'  Others  again  are 
arbitrarily  shaped  marks  standing  for  simple  words  such 
as  not,  and,  or,  if,  is,  identical  with,  A  third  kind  of 
shorthand  symbol  is  one  introduced  in  the  course  of  a 
symbolic  calculus,  and  defined  in  terms  of  combinations 
of  other  shorthand  symbols,  and  ultimately  in  terms 
of  the  simple  symbols  introduced  at  the  outset.    So  far, 


42 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


43 


a  shorthand  symbol  has  all  the  characteristics  of  a  word 
or  a  word-complex — only  differing  from  these  in  satisfying 
the  essential  symbolic  requirements  of  systematisation, 
brevity  and  exactness.  In  one  respect,  however,  these 
symbols  differ  from  such  word-complexes  as  'that  man/ 
*the  river,'  *Mr  Smith,'  'this  experience,'  'my  present 
purpose,'  in  that  these  latter  have  a  meaning  or  appli- 
cation— not  universally  fixed — but  determined  only  by 
means  of  context ;  whereas  the  symbols  of  Logic  have 
an  unalterable  meaning  wholly  independent  of  context, 
and  resemble  rather,  such  word-complexes  as  'rational 
animal,'  'loud,'  'hard,'  'church,'  differing  from  these 
however  in  being  strictly  unambiguous.  Ordinary  Logic 
generally  dispenses  with  symbols  of  this  kind — the  most 

"-^familiar  exception  being  Dr  Keynes's  'SaP/  which  is 
shorthand  for  'Every  5  is  PJ  etc.    On  the  other  hand, 

"^Dr  Keynes  himself  shows,  in  his  Appendix  C,  how 
certain  complicated  problems,  previously  relegated  to 
Symbolic  Logic,  can  be  solved  without  recourse  to 
shorthand  symbols,  illustrative  symbols  only  being  in- 
troduced. 

Now  an  important  character  of  the  shorthand  symbol 
is  that  its  constancy  is  logical  or  formal  and  not  expe- 
riential or  material.  A  formal  constant  is  one  whose 
meaning  is  to  be  understood  by  the  logician  as  such; 
that  is  to  say,  logic  pronounces  it  either  as  indefinable — 
because  understood  without  requiring  definition — or  as 
definable  in  terms  of  logically  understood  constants 
alone.  The  following  is  a  rough  classification  of  formal 
constants  expressed  inordinary  language :  ( i )  the  articles 
or  applicatives ;  a,  the,  some,  etc.  (2)  the  negative  not\ 
and  the  conjunctions  and,  or,  if,  etc.    (3)  the  copula  is\ 


H 


u 


and  certain  prepositions  such  as  0/,  to,  in  some  of  their 
meanings.  (4)  certain  relations  such  as  identical  with, 
comprised  in.  (5)  such  modal  adjectives  as  true,  false, 
probable,  etc.  Formal  constants  are  to  be  contrasted 
with  material  in  that  the  meanings  of  the  latter  are  to 
be  understood  in  terms  of  ideas  or  conceptions  outside 
the  sphere  of  logic.  The  division  between  formal  and 
material  constants,  i.e.  between  what  is  and  what  is  not 
required  for  the  understanding  of  logical  principles,  can 
ultimately  be  rendered  precise  only  after  a  complete 
logical  system  has  been  constructed.  For  instance, 
numerical  adjectives  such  as  two  andyfz^^  would  have 
been  pronounced  as  merely  material  at  the  stage  at 
which  the  logical  system  had  not  been  carried  on  into 
its  mathematical  developments.  Ideas  that  are  imme- 
diately recognised  as  material  relatively  to  the  essentials 
of  logic  are  those  of  sense-qualities,  or  of  the  properties 
and  characteristics  of  physical  and  mental  entities. 
Temporal  and  spatial  relations,  being  in  one  aspect  sub- 
sumable  under  the  conceptions  of  order,  would,  so  far, 
be  called  formal  or  logical,  but,  inasmuch  as  these  rela- 
tions actually  have  a  specific — over  and  above  their 
generic — significance,  they  must  be  treated  also  as 
having  an  experiential  or  material  source.  The  same 
holds  of  the  determinates  of  a  determinable,  inasmuch 
as  experience  is  required  in  order  to  present  to  the 
mind  any  single  determinable  and  to  distinguish  one 
determinable  from  another,  whilst  the  discussion  of  the 
formal  relations  of  incompatibility,  order,  etc.,  between 
determinates  under  any  determinable  is  purely  logical. 
Since  shorthand  symbols  and  the  words  or  word- 
complexes  of  ordinary  language  function  in  the  same 


^1 


44 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


45 


way,  there  is  no  essential  difference  between  them  if 
we  take  the  symbols  or  words  in  isolation  apart  from 
consideration  of  the  mental  processes  involved  in  their 
The   psychological    distinction  is — not  between 


use. 


words  and  symbols  as  such — but  between  the  linguistic 
and  the  symbolic  mode  in  which  we  think  with  their 
assistance.  Thus,  in  linguistic  thought,  the  words  or 
symbols  presented  in  imagination  or  vocalisation  are  the 
means  or  instruments  by  which  we  can  attend  to  or 
think  about  the  objects  for  which  they  stand.  On  the 
other  hand,  such  a  phrase  as  *  Waterloo  was  fought  in 
1815'  might  illustrate  the  symbolic  use  of  language 
which  consists — not  in  thinking  about  the  objects  for 
which  the  words  stand — but  in  mentally  rehearsing  the 
language  in  which  propositions  previously  accepted  have 
been  expressed.  Now  the  previous  acceptance  of  these 
propositions  must  have  entailed  genuine  processes  of 
thinking;  but,  when  they  are  recalled,  we  need  not 
repeat  these  mental  processes.  It  is  in  this  way  that 
the  symbolic  is  distinguished  from  the  linguistic  use  of 
words  or  symbols.  In  the  latter,  we  are  thinking  by  the 
use  of  words;  whereas,  in  the  former,  recall  of  the 
words  serves  merely  as  a  substitute  for  a  previous  act 
of  thought^ 

§  2.  These  preliminary  considerations  bring  us  to 
the  question :  What  actually  happens  in  the  mind  of  the 
symbolist,  when  he  is  either  constructing  or  intelligently 
following  the  formulae  of  a  symbolic  calculus?  In  the 
first  place,  the  axioms  of  the  calculus  can  only  be  es- 
tablished by  the  use  of  what  I  have  called  the  Counter- 

^  This  subject  will  be  found  to  be  more  fully  treated  in  Dr  Stout's 
Analytic  Psychology. 


u 


1, 


i 


■^ 


applicative  and  Counter-implicative  principles,  and  here 
genuine  thought  is  required  on  the  part  of  the  symbolist. 
In  the  second  place,  the  construction  of  any  symbolic 
calculus  involves  the  procedure  of  inference;  and  this 
is  conducted  always  in  accordance  with  the  Applicative 
principle,  and,  in  the  case  of  the  logical  calculus,  also  in 
accordance  with  the  Implicative  principle.  When  pro- 
ceeding in  accordance  with  these  principles,  the  sym- 
bolist is  actually  thinking ;  he  is  not  merely  recalling 
verbal  formulae  in  which  the  results  of  previous  acts  of 
thought  have  been  expressed.  In  the  third  place,  even 
a  perfectly  constructed  symbolic  system  would  need  to 
introduce  some  axioms,  as  also  some  propositions  derived 
from  axioms,  that  can  only  be  expressed  in  non-symbolic 
terms.  This  necessary  recourse  to  ordinary  language  in 
developing  a  deductive  system  shows  that  direct  atten- 
tion to  meanings,  presented  linguistically,  is  entailed  in 
the  intelligent  following  of  even  a  professedly  symbolic 
exposition.  Lastly,  the  extent  to  which  thought  can  be 
dispensed  with,  when  working  a  calculus,  depends  very 
largely  and  essentially  upon  the  extent  to  which  the 
system  requires  what  maybe  called  interpretation  clauses 
such  as  'when  P  stands  for  any  proposition,'  or  'where 
X  is  to  be  understood  as  a  variable  and  ti  as  a  constant.* 
If  the  symbolic  language  is  so  constructed  that  a  mini- 
mum of  interpretation  clauses  is  required,  then  there  is 
a  corresponding  minimum  in  the  extent  to  which  actual 
thinking  is  involved.  But,  however  few  interpretation 
clauses  are  required,  the  intelligent  use  of  symbolic 
formuke  cannot  be  reduced  to  a  merely  mechanical 
process.  This  will  be  still  more  apparent  from  an  exami- 
nation of  the  nature  of  a  symbolic  system  in  which  both 


46 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


47 


shorthand  and  illustrative  symbols  enter  in  combination 
with  one  another. 

§  3.  For  this  purpose  we  will  further  consider  the 
characteristics  of  illustrative  symbols.  These,  being 
nothing  but  arbitrarily  chosen  letters  of  the  alphabet, 
differ  from  words  of  ordinary  language  in  that  they 
cannot  be  interpreted  as  standing  for  this  rather  than 
for  that  specific  object  or  idea;  and  hence,  in  the  nature 
of  the  case,  have  a  variable  application.  The  writer  or 
reader  of  a  symbolic  system  must  always  bear  in  mind, 
however,  that  the  variability  in  application  of  an  illus- 
trative symbol  in  any  given  case  is  not  wholly  unre- 
stricted, but  is  limited  within  an  understood  range. 
Thus  a  single  letter  used  illustratively  must  be  under- 
stood to  be  restricted  in  one  case  (say)  to  any  substan- 
tive; in  another  (say)  to  any  adjective;  and  in  another 
again  (say)  to  any  proposition, — these  being  the  three 
most  prominent  categories  to  which  illustrative  symbols 
are  applied.  Symbolic  devices  may,  indeed,  be  invented 
by  which  to  distinguish  one  kind  of  symbol  as  appli- 
cable to  a  substantive,  and  another  kind  to  some  other 
specific  category;  but  the  range  of  application  to  be 
understood  by  letters  taken  in  combination  could  not  be 
indicated  by  any  such  device.  When  single  letters  are 
bound  together  into  a  complex  by  means  of  logical 
constants,  then  a  further  act  of  intelligence  is  required 
in  interpreting  such  complex.  For  example,  under- 
standing in  the  first  place  the  letters  /,  q,  r,  to  stand 
for  propositions,  such  constructs  as  'p  and  q,'  'p  or  q' 
'p  if  q,'  must  h^  further  interpreted  as  also  constituting 
propositions.  Thus,  when  a  formula  about  any  or  all 
propositions  has  been  established,  we  may  proceed  to 


I 


s 


apply  it  to  any  complex  such  as  '/  and  q^  ox  'pxi  q'  and 
so  on  in  accordance  with  the  Applicative  principle,  in- 
asmuch as  each  of  such  complexes  constitutes  a  pro- 
position.   Similarly,   when  such  letters  as  x,  s,  t  are 
understood  to  stand  for  substantives,  and  such  letters 
as/,  q,  r  to  stand  for  adjectives,  then  a  further  act  of 
intelligence  is  required  to  interpret  such  a  complex  as 
*j  IS p'  as  standing  for  a  proposition.    This  presupposes 
that  the  logical  analysis  of  the  simple  proposition  into 
the  form  *s  is/,'  where  s  is  understood  to  stand  for  a 
substantive  and  p  for  an  adjective,  has  been  discussed 
and  established  in  a  preliminary  account  in  which  words 
and  not  symbols  were  employed.    Propositional  signifi- 
cance having  been  attached  to  this  form  of  construct,  a 
distinct  act  of  intelligence  is  required  when,  in  uniting 
say  ' s  is/'  with  7  is  q'  in  some  form  of  combination, 
the  resulting  construct  is  understood  to  stand  for  a  pro- 
position.   As  another  example  illustrating  the  need  for 
intelligent  activity  in  symbolic  work,  we  may  take  the 
two  propositional  forms  's  e p'  and  'x  i y^  where  *^*  is 
shorthand  for  the  copula  'is'  and  'V  for  is  identical 
with.    Not  only  must  these  forms  be  interpreted  as 
standing  for  propositions,  but  the  relation  for  which  V 
stands  must  be  understood  to  be  different  from  that  for 
which  'e'  stands.    In  consequence,  when  these  two  forms 
occur,  reference  must  be  made  to  one  set  of  established 
formulae  for  the  one  case,  and  to  a  different  set  for  the 
other.    The  necessity  for  using  this  modicum  of  intelli- 
gence is  to  be  contrasted   with   the  purely  blind  or 
mechanical  process  required  of  the  reader  or  writer  in 
making  use  of  the  formulae  to  which  he  refers;  for,  in 
this  latter  process,  he  need  attach  no  significance  to  'e' 


48 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


49 


or  to  *z7  as  each  standing  for  its  own  specific  relation. 
The  examples  adduced  have  been  selected  on  the  ground 
of  their  simplicity,  but  complex  examples  would  have 
brought  out  more  forcibly  the  importance  of  the  distinc- 
tion between  the  intelligent  and  the  merely  mechanical 
operations  required  in  working  a  symbolic  system. 

§  4.  Now  the  variability  that  characterises  illustra- 
tive symbols  constitutes  a  special  feature  of  symbolism, 
and  its  further  discussion  requires  the  introduction  of 
the  notion  function.  This  term  is  used  by  logicians 
and  mathematicians  in  a  sense  quite  unconnected  with 
the  biological  meaning  of  the  term.  The  notion  of  a 
function  is  closely  connected  with  the  notion  of  a  con- 
struct, but  the  former  must  be  understood  relationalfyy 
whereas  this  is  not  obviously  the  ease  with  the  term 
construct.  Thus,  we  should  speak  of  a  certain  construct 
as  being  a  function  of  certain  enumerated  constituents. 
The  notation  for  a  function  in  general  is/(^,  ^,  c,  ...) 
where  a^  b,  c,  ...  stand  for  the  constituents ;  and  where 
the  order  in  which  these  constituents  are  written  is 
essential,  so  that/(^,  b,  c,  ...)  is  not  necessarily  equiva- 
lent to/(^,  ^,  <J,  ...).  Thus  any  function  of  ^,  /J,  r,  ...  is  a 
construct  involving  a,b,c,  ....  But,  if  this  were  all  that 
could  be  said  about  a  function,  the  term  would  have  no 
special  value,  since  it  would  be  a  mere  synonym  for 
*  construct  involving.'  The  importance  of  the  notion 
of  function  lies  in  the  fact  that  we  may  speak  of  the 
same  function  in  reference  to  different  constituents, 
whereas  the  same  construct  would  of  course  entail  the 
same  constituents.  Thus,  if  C  be  a  certain  construct 
involving  a,  b,  r,  ...,  and  if  D  be  another  construct 
involving  /,  q,  r,  ...,  then  C  is  said  to  be  the  same 


*  \ 


,< 


function  of  a,  b,  c,  ...  as  is  /?  of /,  ^,  r,  ...,  when  the 
substitution  of/  for  a,  q  for  b,  r  for  c,  etc.,  would  render 
D  identical  with  C  Thus,  in  order  to  decide  as  regards 
two  constructs,  whether  they  express  the  same  or  a 
different  function,  we  must  specify  the  constituents  of 
which  the  construct  is  regarded  as  a  function  ;  and,  to 
avoid  all  possible  ambiguity,  all  the  constituents  for 
which  substitutions  have  to  be  made  must  be  enu- 
merated. To  explain  this  necessity,  it  must  be  pointed 
out  that  a  construct  may  involve,  implicitly  or  explicitly, 
other  constituents  in  addition  to  those  ^  which  it  is  to 
be  regarded  as  a  function.  In  order  to  indicate  the 
sameness  of  function  exhibited  by  different  constructs, 
it  is  therefore  essential  to  enumerate  those  constituents 
for  which  substitutions  are  contemplated.  These  con- 
stituents will  be  called  variants^ ^  because  it  is  these  and 
these  alone  that  have  to  be  varied  in  order  to  obtain 
the  different  constructs  that  exhibit  the  same  function. 
On  the  other  hand,  in  exhibiting  identity  of  function, 
terms  entering  into  the  construct  that  are  not  to  be 
replaced  by  some  other  terms  will  be  called  constants  or 
non- variants.  Hence  the  distinction  between  a  variant 
and  a  non-variant  constituent  of  a  construct  has  rele- 
vance only  to  functional  identity.  Since  a  function  and 
its  variants  are  to  be  understood  relationally  to  one 
another,  we  may  speak  of  the  variants  for  a  certain 
function  just  as  we  speak  of  a   function  of  certain 

^  The  word  variant  is  here  and  throughout  used  in  place  of  the 
mathematically  technical  word  argument^  partly  in  order  to  prevent 
confusion  with  the  ordinary  logical  use  of  the  latter  word,  and  partly 
in  order  to  bring  out  the  distinction  and  connection  between  the 
notion  of  variant  and  that  of  variable, 

J.  L.  H  4 


50 


CHAPTER  III 


variants.  In  a  complicated  symbolic  system  it  is  found  to 
be  convenient  to  use,  in  place  of  a  singular  or  proper 
name,  an  illustrative  symbol — which,  qua  symbol,  must 
be  what  is  called  variable.  Variability  is  therefore  the 
mark  of  an  illustrative  symbol  as  such,  whereas  the  con- 
trast between  variant  and  non-variant  holds — not  of  a 
symbol — but  of  that  for  which  the  symbol  may  stand  ; 
and,  as  has  been  said,  this  latter  contrast  has  no  sig- 
nificance apart  from  the  notion  of  a  function. 

§  5.  In  considering  the  constituents  of  a  construct 
with  a  view  of  indicating  which  are  to  be  variants  and 
which  non-variants  for  a  function,  we  must  first  note 
the  distinction  between  material  and  formal  constituents. 
Now  as  regards  the  strictly  formal  constituents  of  a 
construct,  logic  never  contemplates  making  substitu- 
tions for  these ;  hence,  in  all  applications  of  the  notion 
of  a  function  in  reference  to  its  variants,  two  cases 
only  have  to  be  considered  ;  (i)  the  function  for  which 
all  the  material  constituents  are  treated  as  variants, 
and  (2)  the  function  for  which  some  of  the  material 
constituents  are  treated  as  constants  and  others  as 
variants — in  both  cases  the  formal  constituents  being 
understood  to  be  constants.  When  (i)  all  the  material 
constituents  are  to  be  varied,  then  the  function  may  be 
said  to  be  formal ;  and  the  form  of  a  construct  is  a 
brief  synonym  for  the  formal  function  which  it  exhibits. 
But,  when  (2)  some  of  the  material  constituents  are  to 
be  constant,  then  the  function  will  be  said  to'be  non- 
formal.  It  follows  that,  when  two  constructs  can  be 
said  to  exhibit  the  same  formal  function,  their  reduction 
to  identity  is  effected  by  taking  all  the  formal  con- 
stituents to  be  constant,  and  replacing  all  the  material 


1 


SYMBOLISM  AND  FUNCTIONS  51 

constituents  of  the  one  by  those  of  the  other.  But, 
when  two  constructs  are  said  to  exhibit  the  same  non- 
formal  function,  their  reduction  to  identity  is  effected 
by  taking  certain  of  the  material,  as  well  as  all  the 
formal,  constituents  to  be  constant,  and  replacing  all 
the  remaining  material  constituents  of  the  one  by  those 
of  the  other.  A  formal  function  is  a  function  of  all  the 
material  constituents,  since  all  these  are  to  be  varied  ; 
but  a  non-formal  function  is  a  function  of  only  some  of 
the  material  constituents,  because  only  some  of  these 
are  to  be  varied. 

We  may  take  the  following  as  illustrations  of  formal 
functions  :  The  construct  '  a  good  boy '  is  the  same 
function  of  the  variants  good  and  boy  as  is  *  a  diffi- 
cult problem '  of  the  variants  difficult  and  problem  ; 

*  Socrates  is  wise  *  is  the  same  function  of  Socrates  and 
wise,  as  is  '  London  is  populous '  of  London  and  popu- 
lous ;  *  red  or  heavy '  is  the  same  function  of  red  and 
heavy  as  is  '  loud  or  pleasant '  of  loud  and  pleasant. 
We  may  compare  these  simple  examples  with  similarly 
simple  examples  in  arithmetic.  The  arithmetical  con- 
struct *  three  days  plus  seven  days'  is  the  same  function 
of  the  two  variants  three  days  and  seven  days  as  is 
'  five  feet  plus  four  feet '  of  the  two  variants  five  feet 
and  four  feet ;  '  four  days  multiplied  by  three '  is  the 
same  function  of  four  days  and  three  as  is  '  seven  feet 
multiplied  by  two '  of  seven  feet  and  two,  etc.  These 
illustrate  formal  functions  because  the  only  constituents 
which  are  constant  are  formal:  namely  'a,'  *is,'  'or,' 

*  plus,'  '  multiplied  by,'  respectively.  Each  of  the  above 
examples  exhibits  a  specific  formal  function,  and  serves 
to  explain  the  general  notion  of  a  formal  function.  We 

4—2 


52 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


53 


may  take  similar  examples  to    illustrate  the   general 
notion  of  a  non-formal  function.    Thus  taking  boy  as 
constant,  *  a  good  boy  '  is  the  same  function  of  good  as 
is  'a  tall  boy'  oi  tall\  taking ^^^^  as  constant,  'a  good 
boy'  is  the  same  function  of  boy  as  is  'a  good  action 
of  action]  taking p/easant  a.s  constant,  'loud  or  pleasant 
IS  the  same  function  of  loud  as  is  *  bright  or  pleasant 
of  bright ;  taking  wise  as  constant,  '  Socrates  is  wise 
is  the  same  function  of  Socrates  as  is  *  Plato  is  wise 
of  Plato  ;    taking  Socrates  as  constant,    '  Socrates  is 
wise '  is  the  same  function  of  wise  as  is  '  Socrates  is 
poor'  oi poor,  etc.,  etc.    And  in  general  the  specific 
function  exhibited  by  a  given  construct  varies  according 
to  the  constituents  of  the  construct  that  operate  as 
variants  \ 


^  It  will  be  observed  that  in  the  above  illustrations  of  non-formal 
functions  we  have  used  adjectives  and  substantives  indifferently  as 
constants  or  as  variables.  Now  in  Mr  Russell's  first  introduction  of 
the  notion  of  function,  he  appears  to  limit  the  application  of  the 
notion  to  the  case  where  the  substantive  is  variable  and  the  adjective 
is  constant.  It  is  true  that  he  extends  the  notion  to  include  the  cases 
in  which  the  reverse  holds;  yet  throughout  he  adopts  an  absolute 
distinction  between  the  two  constituents  of  a  proposition  which  I 
have  called  substantive  and  adjective,  inasmuch  as  he  treats  the  sub- 
stantive as  the  typical  kind  of  entity  which  can  stand  by  itself,  the 
adjective  never  being  allowed  to  stand  by  itself.  Thus  I  am  repeating 
his  illustration  in  giving  'Socrates  is  wise'  as  the  same  function  of 
Socrates  as  is  '  Plato  is  wise '  of  Plato,  since  here  the  substantive  terms 
Socrates  and  Plato  are  allowed  to  stand  by  themselves.  But  the 
parallel  example,  that  *  Socrates  is  wise '  is  the  same  function  of  wise 
as  is  *  Socrates  is  poor '  oi poor,  is  not  recognised  by  Mr  Russell,  be- 
cause he  does  not  allow  such  adjective-terms  as  *  wise '  and  *  poor '  to 
stand  by  themselves.  The  consequences  of  this  contrast,  which  I  hold 
to  be  fundamentally  fallacious,  between  the  substantive  and  the 
adjective  as  constituents  of  a  proposition,  infect  the  whole  of  his  logical 


^ 


\ 


§  6.  A  classification  has  been  given,  in  an  earlier 
section,  of  those  formal  constituents  of  a  construct  that 
are  expressible  in  words  or  in  shorthand  symbols  under- 
stood as  equivalent  to  words.  Such  formal  constituents 
may  be  called  explicit  in  distinction  from  others  which 
are  more  or  less  latent  and  not  usually  expressed  in 
words.  Reserving  the  name  '  constituent,'  for  the 
material  variants,  and  'formal  component'  for  those 
formal  constants  that  are  explicitly  expressed,  the 
implicit  formal  constants  may  be  conveniently  termed 
*  elements  of  form.'  Of  these,  several  different  kinds 
are  to  be  distinguished  : 

(i)  Ties.  These  are  more  or  less  latent  elements  of 
form,  inasmuch  as  it  is  a  matter  of  accident  whether  they 
are  expressed  by  some  separate  word  or  by  some  form 
of  grammatical  inflection. 

(2)  Brackets,  A  construct  may  be  composed  of 
sub-constructs,  and  these  again  of  sub-sub-constructs 

system.  Without  entering  into  elaborate  detail,  it  would  be  impossible 
fully  to  justify  my  difference  from  Mr  Russell  on  this  matter ;  but 
what  I  take  to  be  perhaps  the  root  of  the  error  is  that  he  treats  the 
general  notion  of  function  before  giving  examples  of  the  simplest 
functional  forms  upon  which  the  more  complicated  functions  are  built. 
It  is  true  that  he  illustrates  a  function  by  such  an  elementary  example 
as  *  :x;  is  a  man '  where  x  stands  indifferently  for  Socrates  or  Plato,  etc., 
but  he  does  not  bring  out  the  speciality  of  this  form  of  proposition, 
which  does  in  fact  exhibit  the  specific  function  which  is  constructed 
by  means  of  the  copula  *  is.'  In  mathematics  the  general  notion  of 
function  is  reached  by  building  up  constructs  out  of  such  elementary 
functions  as  those  indicated  by  +  x  -  etc.,  but  in  Mr  Russell's 
system  it  seems  impossible  to  explain  and  reduce  to  systematic 
symbolisation  the  process  by  which  any  prepositional  function  what- 
ever is  constructed. 

I  hope  to  treat  more  fully  elsewhere  this  point  of  difference  be- 
tween Mr  Russell's  system  and  my  own. 


/ 


54 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


55 


and  so  on,  until  we  reach  the  ultimate  constituents, 
namely  those  that  are  expressed,  not  as  constructs,  but 
as  'simples/  where  by  'simple'  is  not  meant  incapable 
of  analysis,  but  merely  unanalysed.  The  operation  of 
binding  constituents  into  a  unity  to  constitute  a  sub- 
construct  I  shall  call  bracketing.  In  speaking,  the  dis- 
tribution of  brackets  is  indicated  by  pauses  or  vocal 
inflections ;  and,  in  writing,  by  punctuation  marks. 
But,  as  the  employment  of  these  signs  is  not  governed 
by  any  systematic  principle,  they  must  be  replaced  in 
logical  or  mathematical  symbolism  by  some  conven- 
tional notation. 

(3)  Connectedness.  Two  sub- constructs  will  be  called 
unconnected  when  one  is  a  function  of  the  simple  terms 
a,  b,  c  (say),  and  the  other  of  the  simple  terms  d,  e — 
the  terms  of  the  one  not  recurring  in  the  other.  On  the 
other  hand,  two  sub-constructs  will  be  called  connected 
when  one  is  a  function  of  a,  b,  c  (say),  and  the  other  of 
a,  e — the  term  a  recurring  in  the  two.  This  distinction 
is  of  importance  when  we  have  to  determine  what  con- 
stituents of  a  function  can  be  taken  as  variants ;  for 
the  several  variants  for  a  function  must  be  indepen- 
dently variable,  and  in  the  case  of  any  two  complex 
constituents,  if  these  are  connected  (in  the  sense  ex- 
plained), they  cannot  be  made  to  vary  independently 
the  one  of  the  other.  Thus,  in  the  above  illustration  of 
two  sub-constructs  that  are  respectively  functions  of 
a,  b,  c  and  of  ^,  e,  the  variants  for  the  function  exhibited 
by  the  construct  must  be  taken  to  be  the  'simple' 
constituents  a,  b,  c,  e,  and  not  the  connected  sub- 
constructs  themselves.  But,  when  a  construct  contains 
unconnected  sub-constructs,  as  in  the  example  of  the 


T 


t 


^\- 


^ 


sub-constructs  that  are  functions  respectively  of  a,  b,  c 
and  of  ^,  ey  then  it  may  be  regarded  either  as  a  function  of 
the  several  ultimate  terms  involved  in  the  different  sub- 
constructs,  namely  a,  3,  c,  d,  e\  or  alternatively,  as  a 
function  of  the  sub-constructs  themselves. 

(4)  Categories.  Every  material,  and  therefore  vari- 
able, constituent  belongs  to  a  specific  logical  category 
or  sub-category  which  is  not  usually  expressed  in  words. 
Thus  the  proposition  '  Socrates  is  wise  '  is  understood 
as  it  stands  without  being  expanded  into  the  form 
'  The  substantive  Socrates  is  characterised  by  the 
adjective  wise.'  Nevertheless  the  formal  significance 
of  the  proposition  for  the  thinker  depends  upon  his 
conceiving  of  '  Socrates '  as  belonging  to  the  category 
substantive,  and  of  'wise'  as  belonging  to  the  category 
adjective.  These  must  therefore  be  included  amongst 
the  latent  elements  of  form.  It  further  follows  from 
the  recognition  of  this  formal  element,  latent  in  every 
material  constituent,  that  the  range  of  variation  for 
any  material  constituent  is  determined  by  the  logical 
category — substantive,  adjective,  relational  adjective, 
as  the  case  may  be — to  which  it  belongs.  In  other 
words,  the  material  constituents  which  may  replace  one 
another,  in  order  that  the  construct  may  exhibit  the 
same  function  in  its  varied  exemplifications,  must  all 
belong  to  the  same  logical  category  or  sub-category. 

§  7.  This  account  of  the  formal  elements  of  a  con- 
struct leads  to  an  examination  of  different  types  of 
function.  Amongst  the  functions  of  logic  the  con- 
junctional and  the  predicational  are  the  most  funda- 
mental. A  function  is  called  conjunctional  when  the 
component  that  determines  its  form  is  the  negative  not 


< 


56 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


57 


or  some  logical  conjunction ;  and  the  variants  for  such 
a  function  are  always,  strictly  speaking,  propositions, 
as  is  also  the  construct  itself.    A  function  is  called 
predicational  when  the  component  that  determines  its 
form  is  the  characterising  tie,  which  unites  two  variants 
related  to  one  another  as  substantive  to  adjective.   Thus 
there  is  only  one  elementary  predicational  function, 
namely  the  characterising  function  represented  by  the 
copula  *is';  whereas  there  are  five  elementary  conjunc- 
tional functions  represented  respectively  by  the  opera- 
tors, 'not,'  *and,'  *if*  and  its  converse,  *or/  *not-both.' 
Just  as  a  conjunctional  function  may  exhibit  any  degree 
of  complexity  made  up  of  these  elementary  conjunctional 
functions,  so  a  predicational  function  may  exhibit  any 
degree  of  complexity  made  up  of  recurrences  of  the 
characterising  function  in  sub-constructs  and  sub-sub- 
constructs,  etc.    An  important  distinction  between  these 
two  types  of  function  introduces  the  notion  of  func- 
tional homogeneity.    A  function  is  said  to  be  homo- 
geneous  when    all    its  variants   belong    to   the   same 
category  as  itself.    Now,  since  a  conjunctional  function 
takes  propositions  as  its  variants  and  is  itself  a  pro- 
position,  it  illustrates  a  homogeneous  function ;  but, 
since  a  predicational  function  constitutes  a  construct 
under  the  category  proposition  out  of  constituents  under 
the  respective  categories,  substantive  and  adjective,  it 
illustrates  a  heterogeneous  function.    Under  this  head 
are  also  to  be  included  secondary  propositions  which 
predicate  adjectives  of  primary  propositions,  and  pro- 
positions which  predicate  secondary  adjectives  of  pri- 
mary adjectives;  for  the  subjects  of  these  propositions 
are  quasi-substantives,  and  the  propositions  themselves 


<v 


1 

i 
4 


are  of  a  different  order  of  category  from  their  con- 
stituent terms. 

§  8.  We  will  proceed  to  apply  the  notion  of  con- 
nectedness to  these  two  types  of  function.  A  conjunc- 
tional function  is  a  function  of  those  propositional  sub- 
constructs  which  are  unconnected,  but  not  of  those 
which  are  connected  with  one  another  through  identity 
of  some  of  the  terms  involved.  For  such  sub-constructs, 
though  properly  regarded  as  constituents,  cannot  be 
taken  as  variants,  since  they  cannot  be  freely  varied 
independently  of  one  another.  Thus  the  variants  for  a 
conjunctional  function  which  is  also  connectional  are 
not  the  connected  sub-constructs  themselves,  but  the 
ultimate  propositions  or  *  simples*  of  which  they  are 
constituted;  e.g.  in  the  construct 

{(/  and  q)  or  (^  and  r)}  and  {^x  or  y) 

the   constituents  that  may  be  taken  as  variants  are 
/,  q,  r,  {x  ox y)\  and  in  the  construct 

{(/  and  q)  or  (^  and  r)}  and  (^  or^) 

the  only  constituents  that  can  be  taken  as  variants  are 
/>  Qi  ^»  y*  I^  these  symbolic  illustrations,  the  ultimate 
constituents  are  unanalysed  propositions ;  but  the  same 
distinction  between  connected  and  unconnected  sub- 
constructs  holds  for  a  conjunctional  function  of  pro- 
positions that  are  expressed  analytically  in  terms  of 
subject  and  predicate.  For  example,  'A  \s p  or  B  \s  q' 
illustrates  a  conjunctional  function  of  the  two  uncon- 
nected sub-constructs  '  A  \s p,'  ' B  \s  q,'  On  the  other 
hand,  ^A  is  p  or  A  is  q*  is  not  a  function  of  the  sub- 
constructs  'A  is  p/  'A  is  q'  because  these  are  con- 
nected; but  must  be  taken  as  a  function  of  the  three 


58 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


59 


ultimate  constituents  Ayp,  q.  Again,  'A  isp  or  B  \sp' 
is  not  a  function  o(  'A  \s  p,'  'Bis  p,'  but  of  the  ultimate 
constituents  Ay  B,  p.  The  connectedness  in  the  former 
case  is  through  identity  of  the  substantive  A ;  in  the 
latter  through  identity  of  the  adjective  /.  Similar 
examples  of  connectedness  occur,  in  which  *if'  or  *not- 
both'  or  *and'  enter  in  the  place  of  'or.' 

Ordinary  language  adopts  abbreviated  expressions 
for  propositions  that  are  connected,  through  identity  of 
subject,  by  constructing  a  compound  predicate,  e.g.  'A 
\s p  or  q,'  'A  IS p  and  q'  ]  as  also  for  propositions  that 
are  connected,  through  identity  oi  predicate,  by  con- 
structing a  compound  subject,  e.g.  'A  or  B  is  p,  'A 
and  B  are/.'  This  is  extended  to  any  number  of  terms 
enumeratively  assigned  for  which  language  supplies  us 
with  a  special  condensed  mode  of  expression.  Thus  the 
alternative  function  is  condensed  into  the  form  :  'Some 
one  or  other  of  the  enumerated  items  is  /';  and  the 
conjunctive  function  into  the  form:  '  Every  one  of  the 
enumerated  items  is/.'  Such  forms  are  usually  restricted 
to  enumerations  of  substantival  items  :  for  example, 
*  Some  one  of  the  apostles  was  a  traitor,'  '  Every  one  of 
the  apostles  was  a  Jew.'  But  it  is  possible  to  extend 
the  form  to  enumerations  of  propositional  or  of  adjec- 
tival items ;  for  example,  '  Some  one  of  the  axioms  of 
Euclid  is  unnecessary  for  the  purpose  of  establishing 
the  theorems  of  geometry ' ;  or  '  Every  one  of  the 
qualities  characterising  A,  B,  C  characterises  DJ 

§  9.  A  special  notation  has  been  adopted  by  the 
symbolists  for  representing  such  condensed  expressions. 
In  this  notation,  an  illustrative  symbol  such  as  x  enters 
as  an  apparent  variable  (to  use  Peano's  phraseology) ; 


\' 


I 
t 


by  which  is  meant  that  the  proposition  in  which  x 
occurs — though  it  appears  to  be,  yet  is  not  in  reality — 
about  Xy  inasmuch  as  its  content  is  not  changed  when 
any  other  symbol,  say  y,  is  substituted  for  :*:.  The 
typical  mode  of  formulating  propositions  on  this  prin- 
ciple is  :  '  Every  item,  say  x,  is  /,'  or  *  Some  item,  say 
Xy  is  /,'  where  it  is  obvious  that  the  force  of  the  pro- 
position would  be  unaltered  if  we  substituted  s,  or  y, 
or  z,  for  X.  If  X  is  the  name  of  the  class  that  comprises 
all  such  items  as  x,  then  the  above  forms  are  equi- 
valent to  '  Every  X  is  /,'  and  '  Some  X  is  p'  respec- 
tively. The  ultimate  constituents  of  such  universal  or 
particular  propositions  are  the  simple  propositions  of 
the  form  ' x  isp'  which  are  conjunctively  combined  for 
the  universal,  and  alternatively  combined  for  the  par- 
ticular. The  phrases  *  Every  X,  '  Some  X,  therefore, 
though  obviously  constituents  of  the  sentence,  do  not 
denote  genuine  constituents  of  the  proposition  of  which 
the  sentence  is  the  verbal  expression.  Since  then  the 
constituents  of  the  general  proposition  are  singular 
propositions  of  the  form  'xispj  such  a  class-name  as  X 
and  such  a  variable  name  as  x,  which  are  in  danger 
of  being  identified,  must  be  carefully  distinguished.  To 
the  former  the  distributives  some  or  every  can  be  pre- 
fixed, never  to  the  latter.    [See  Part  I,  Chapter  VII.] 

When  we  use  a  symbolic  variable  or  illustrative 
symbol  x  to  construct  the  proposition  ^x  is  /'  say, 
X  stands,  not  for  a  class-name,  but  for  a  special 
kind  of  singular  name,  only  differing  from  the  ordinary 
singular  name  in  that  it  stands  indifferently  for  any 
substantive  name,  such  as  'Socrates'  or  'Cromwell'  or 
*this  table'  or  'yonder  chair.'    To  bring  out  more  pre- 


II 


6o 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


6i 


cisely  the  distinction  between  a  symbolic  variable  and 
a  class-name,  we  may  suppose  that  in  a  certain  context 
s  stands  indifferently  for  dniy person  such  as  'Socrates* 
or  'Cromwell';  or  again,  indifferently  for  any  article 
of  furniture  such  as  'this  table'  or  'yonder  chair/ 
Now  'person'  and  'article  of  furniture'  are  class-names, 
and  in  the  instances  adduced  the  symbolic  variable  s 
stands — not  for  the  class-name — but  in  fact  for  any 
singular  name  (proper  or  descriptive)  that  denotes  an 
individual  comprised  in  the  class  '  person '  for  the  one 
case,  and  the  class  'article  of  furniture'  for  the  other  case. 
What  holds  of  a  substantive-name  s  holds  also  of  an 
adjective-name  /  or  of  a  class-name  c.  Thus,  in 
the  form  's  is  /,'  where  'is'  represents  the  charac- 
terising tie,  p  stands  for  any  one  indifferenriy  assign- 
able adjective  comprised,  say,  in  the  class  colour,  but 
not  for  the  class  itself  to  which  the  distributives 
'every'  or  'some'  can  be  prefixed.  Again,  in  the 
form  's  is  comprised  in  c^  c  represents  a  singular 
class-name  standing  for  any  one  indifferently  assignable 
class;  and  the  limits  of  variation  for  the  variable  c 
could  be  expressed  in  terms  of  a  class  of  a  higher 
order  comprising  it.  Thus  the  symbol  c  is  equivalent  to 
a  variable  proper  class-name,  and,  like  the  substantive- 
name  s  and  the  adjective-name  /,  is  to  be  contrasted 
with  the  class  in  which  it  is  comprised.  The  names 
substantive,  adjective,  proposition,  etc.,  which  denote 
logical  categories,  i.e.  the  ultimate  comprising  classes, 
are  not  variable  proper  names,  but  names  bearing  fixed 
or  constant  significance,  having  so  far  the  character  of 
shorthand  symbols  in  that  they  stand  for  logical  con- 
stants, not  for  material  variables.    Thus  the  employ- 


I 


^• 


ment  of  the  illustrative  symbol  as  an  apparent  variable 
— i.e.  to  stand  indifferently  for  any  one  or  another 
object — makes  possible  the  use  of  the  same  symbol, 
recurring  in  a  given  context,  to  stand  for  the  same 
object.  It  thus  fulfils  the  same  function  in  a  complex 
symbolic  formula  as  the  proper  name  in  ordinary  narra- 
tive, where  the  use  of  the  pronoun  in  complicated  cases 
would  be  ambiguous.  The  construction  of  such  formulae 
requires  the  use,  in  a  symbolic  system,  of  apparent  vari- 
ables in  place  of  class-names. 

§  ID.  We  have  seen  that  certain  phrases  containing, 
implicitly  or  explicitly,  the  conjunctions  and  or  or,  though 
linguistically  intelligible,  do  not  really  represent  genuine 
constructs.  This  raises  a  wider  and  more  fundamental 
problem  in  regard  to  the  nature  of  logical  conjunctions 
when  used  in  constructingacompound  out  of  enumerated 
items.  Can  conjunctions  serve  to  construct  compound 
substantives  or  compound  adjectives  in  the  same  way 
as  they  operate  in  constructing  compound  propositions  ? 
Now  I  shall  maintain  that  while  the  nature  of  an  adjec- 
tive is  such  that  we  may  properly  construct  a  compound 
adjective  out  of  'simple'  adjectives  just  as  we  may 
construct  a  compound  proposition  out  of  'simple'  pro- 
positions, yet  the  nature  of  any  term  functioning  as  a 
substantive  is  such  that  it  is  impossible  to  construct  a 
genuine  compound  substantive.  Thus  'rational  and 
animated'  represents  a  genuine  conjunctive  adjective, 
since  it  is  equivalent  in  meaning  to  the  simple  adjective 
'human';  and  'one  or  other  of  the  colours  approximat- 
ing to  red'  is  a  genuine  alternative  adjective,  since  it  is 
equivalent  in  meaning  to  the  simple  adjective  'reddish.' 
And  again,  more  generally,  where  no  single  adjectival 


/ 


62 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


63 


word  represents  such  a  conjunction  of  adjectives  as 
*  square  and  heavy,'  'red  or  green,'  these  are  still  to  be 
regarded  as  genuine  adjectival  constructs  on  the  ground 
that  they  agree  in  all  essentially  logical  respects  with 
simple  adjectives,  from  which  in  fact  they  cannot  be 
distinguished  by  any  universal  criterion.  It  follows, 
therefore,  that  no  contradiction  will  ensue  from  replacing 
a  simple  by  a  compound  adjective  in  any  general  for- 
mula holding  of  all  adjectives  as  such.  At  the  same 
time  it  must  be  pointed  out,  as  regards  alternative  adjec- 
tival constructs,  that  no  single  or  determinate  adjective 
can  be  identified  with  such  an  alternative  or  indetermi- 
nate adjective  as  'red  or  green,'  'one  or  other  of  the 
colours  approximating  to  red.'  In  this  respect,  as  we 
shall  see,  an  alternative  adjectival  construct  precisely 
resembles  a  substantival  construct.  Turning  then  to 
substantival  constructs,  it  is  obvious  in  the  first  place 
that  a  conjunctive  enumeration  of  substantives  such  as 
'Peter  and  James'  or  'Every  one  of  the  apostles'  does  not 
represent  any  single  or  determinate  man.  It  might,  how- 
ever, be  maintained  that  such  phrases  represent  a  couple 
of  men  or  a  class  of  men,  and  that  a  couple  or  a  class 
comprising  substantives  is  itself  of  the  nature  of  a  sub- 
stantive. Such  a  view  would,  however,  involve  a  con- 
fusion between  the  enumerative  and  the  conjunctional 
and.  A  statement  about  'Peter  and  James'  or  'Every 
one  of  the  apostles'  is  really  not  about  the  compound 
construct  that  appears  to  be  denoted  by  its  subject- 
term,  but  must  be  analysed  into  a  conjunctive  c6mpound 
of  singular  propositions.  Thus  in  the  statement  '  Peter 
and  James  were  fishermen'  the  subject-term  uses  and 
enumeratively.    The  conjunctional  and  can  be  shown 


H 


to  enter  only  when  we  analyse  the  statement  into  the 
form  'Peter  was  a  fisherman  and  James  was  a  fisher- 
man.' The  case  of  an  alternative  enumeration  of  sub- 
stantives, such  as  '  Peter  or  James'  or  'Some  one  of  the 
apostles,'  is  less  obvious  than  that  of  a  conjunctive 
enumeration  of  substantives.  To  prove  that  the  alter- 
native enumeration  does  not  represent  a  genuine  sub- 
stantive, it  will  be  convenient  to  take  a  proposition  in 
which  the  enumeration  occurs  in  the  predicate.  Thus 
'Nathaniel  is  one  of  the  apostles'  or  '  Bartholomew  is 
one  of  the  apostles '  would  appear  to  be  expressible  in 
the  form  'Nathaniel  is-identical-with  one  or  other  of 
the  apostles'  or  'Bartholomew  is-identical-with  one  or 
other  of  the  apostles.'  But,  if  this  is  allowed,  the  con- 
junction of  these  two  propositions  would  imply  that 
'Nathaniel  is-identical-with  Bartholomew/  since  things 
that  are  identical  with  the  same  thing  are  identical  with 
one  another.  Now  that  'Nathaniel  is-identical-with 
Bartholomew'  may  or  may  not  be  the  case;  but  it  cer- 
tainly would  not  follow  from  the  fact  that  Nathaniel 
was  one  of  the  apostles  and  that  Bartholomew  was  one 
of  the  apostles.  In  order  correctly  to  formulate  the  pro- 
position 'Nathaniel  was  one  of  the  apostles'  in  terms  of 
the  relation  of  identity,  it  must  be  rendered:  'Nathaniel 
is-identical-with  Peter  or  identical-with  Bartholomew  or 
identical-with  Thaddeus,  etc'  In  this  form,  the  alter- 
nants are  not  the  proper  or  substantival  names  Peter, 
Bartholomew,  Thaddeus,  etc.,  but  the  adjectival  terms 
'identical  with  Peter,'  'identical  with  Bartholomew,' 
identical  with  Thaddeus,'  etc.  These  latter  being  re- 
cognised as  adjectives,  the  reconstructed  proposition 
assumes  the  form  ' A  \s  p  ox  q  or  r,  etc'  where  'is' 


64 


CHAPTER  III 


>   • 


SYMBOLISM  AND  FUNCTIONS 


65 


represents  the  characterising  tie,  and/,  q,  r...  stand 
for  adjectives,  so  that  (as  alleged  above)  the  new 
predicate  expresses  a  genuine  construct. 

§  II.  A  further  and  more  general  explanation  may 
now  be  given  of  the  principle  according  to  which  a 
proposition  containing  a  fictitious  construct  must  be  re- 
formulated. What  holds  of  the  relation  of  identity  (as 
in  the  particular  example  concerning  the  apostles)  holds 
of  any  relation  whatever:  that  is  to  say,  taking  r  to 
stand  for  any  relation,  the  phrase  'r  to  a  or  ^  or  ^...' 
does  not  express  a  genuine  construct  and  must  be  re- 
placed by  the  phrase  ^r  X.o  a  or  r  to  b  or  r\,o  c,,,'  which 
is  an  alternative  of  adjectives.  For  example,  the  pro- 
position *This  action  will  injure  either  Germany  or 
England'  must  be  transformed  into  'This  action  will 
either  injure  Germany  or  injure  England.'  The  essen- 
tial points  in  this  transformation  can  best  be  indicated 
with  the  help  of  vertical  lines  for  brackets.    Thus : 

\^rx,o\aorborc 


X 


is  corrected  into 

x\  is  I  r  to  a  or  r  to  6  or  r  to  c. 

In  the  former  the  two  principal  constituents  of  the  pro- 
position are  linked  by  the  relational  predication  *is  r 
to,'  in  the  latter  by  the  characterising  tie  *is.'  In  order 
that  the  predicate  in  the  latter  case  should  constitute  a 
genuine  construct,  what  is  essential  is,  not  that  the 
subject  term  should  stand  for  a  substantive  in  any  abso- 
lute sense,  but  only  that  it  should  function  as  a  sub- 
stantive relatively  to  the  adjectival  predicate;  and  it  is 
the  characterising  tie  which  indicates  this  relative  con- 
ception of  substantive  to  adjective.    Thus  the  term  x 


4 


i 


\ 


may  be  either  a  substantive  proper,  an  adjective  or  a 
proposition,  and  the  same  holds  of  the  terms  a,  ^,  c, 
with  which  x  is  connected  by  the  relation  r. 

Examples  may  be  given  of  propositions  based  upon 
the  forms  *x  is  r  to  a  or  ^,'  'x  is  r  to  a  and  6,'  in  order 
further  to  illustrate  the  principles  under  discussion. 

(i)  In  the  example  just  given:  'This  action  will 
injure  either  Germany  or  England,'  which  must  be 
rendered  'This  action  will  either  injure  England  or  in- 
jure Germany/  the  terms  Xy  a,  d,  are  all  substantives 
proper.    But  taking 

(2)  'p  characterises  either  a  or  ^  or  ^,'  which  has  to 
be  transformed  into  'p  either  characterises  a  or  charac- 
terises d  or  characterises  ^,'  the  subject  term  is  an  ad- 
jective and  may  be  called  primary  relatively  to  the 
predicate  terms  which  function  as  secondary  adjectives  \ 

In  (3):  'A  has  asserted/  or  ^  or  r,'  the  subject  term 
stands  for  a  person  (i.e.  for  a  substantive  proper),  and 
the  terms/,  q,  r  in  the  predicate  are  propositions.  Since 
here  the  terms  alternatively  combined  are  themselves 
propositions,  the  expression  as  it  stands  would  be  corit  ct 
if  its  intention  were  to  state  that  the  compound  propo- 
sition ' p  or  q  or  r'  was  asserted  by  A,  But,  if  it  were 
intended  to  state  that  one  or  other  of  the  assertions 
/,  q,  r  had  been  made  by  A^  then  (3)  should  be  amended 

^  The  predication  characterises^  like  injures  in  the  previous  example, 
is  expressed  by  a  verb ;  but,  as  explained  in  Part  I,  Chapter  XIIT, 
section  5,  any  verb  may  be  resolved  into  an  adjective  or  relation 
preceded  by  the  characterising  tie.  Thus,  in  order  to  show  more 
explicitly  that  the  principal  constituents  are  united  by  the  characterising 
tie,  proposition  (2)  should  be  expanded  into  the  form:  */  is  charac- 
terised as  either  characterising  a  or  characterising  b  or  characterising 
<:.'   Similarly  for  other  examples. 

J.  L.  n  5 


,4» 


66 


CHAPTER  III 


(as  in  the  preceding  examples)  into  the  form:  'A  has 
asserted/  or  has  asserted  q  or  has  asserted  r' 

(4)  The  proposition :  'g  is  characterised  by  all  the 
adjectives  that  characterise  a  and  b  and  c'  exhibits  a 
higher  degree  of  complexity  than  those  previously  given 
since  it  introduces  the  two  correlatives  characterising 
and  characterised  by.  It  illustrates  a  type  of  proposi- 
tion which  plays  an  important  part  in  the  theory  of 
induction ;  and  is  a  specific  case  of  the  more  general 
form:  'g  is  r  to  everything  that  \^rX.oa  and  b  and  ^.' 
As  thus  formulated  it  contains  the  fictitious  conjunctive 
construct  'a  and  b  and  c'  where  a,  by  c  function  as 
substantives.  To  eliminate  this  fictitious  construct,  the 
statement  must  be  reformulated  thus :  'g  is  character- 
ised by  every  adjective  that  characterises  a  and  charac- 
terises b  and  characterises  c'  But  there  still  remains 
the  fictitious  construct  prefaced  by  the  distributive 
phrase  'every  adjective/  The  final  correction  must  be 
made  by  introducing  an  apparent  variable  as  was  re- 
quired in  reformulating  the  elementary  forms  of  pro- 
position: 'Every  M  is  //  ^Some  M  is  /.'  Thus: 
*  Every  adjective,  say  x,  that  characterises  a  and  charac- 
terises b  and  characterises  c  also  characterises^.' 

§  12.  The  above  exposition  of  functions  is  funda- 
mentally opposed  to  that  given  in  the  Principia  Mathe- 
matica.  The  first  point  of  difference  to  be  emphasised 
concerns  Mr  Russell's  view  of  the  relation  between 
what  he  calls  a  propositional  function,  and  function 
in  the  sense  in  which  it  is  universally  understood  in 
mathematics.  The  latter  he  terms  a  descriptive  function, 
and  maintains  that  it  is  derivable  from  the  nature  of  the 
propositional  function;  whereas  it  appears  to  me  that 


,1 


I 


1 


SYMBOLISM  AND  FUNCTIONS  67 

the  reverse  is  the  case,  and  that  his  propositional  func- 
tion IS  nothing  but  a  particular  case  of  the  mathematical 
Junction.    The  general  nature  of  a  descriptive  function 
can  be  illustrated  by  taking  a  proposition  say  about 
The  teacher  oiy.'  This  phrase  illustrates  what  is  meant 
by  a  descriptive  function,  the  full  meaning  of  which  can 
be  indicated  only  by  showing  how  it  may  enter  into  a 
proposition  such  as  (a) :  '  The  teacher  ofy  was  a  Scotch- 
man.    Now  we  may  agree  with  Mr  Russell  that  this 
proposition  could  not  be  interpreted  as  true,  unless  y 
had  one  and  only  one  teacher.    On  this  interpretation 
the  full  force  of  the  proposition  is  explicated  as  follows: 
(a)    There  is  a  being,  say  6,  of  which  the  following 
statements  may  be  made:  '^ 

(i)   that  d  was  a  Scotchman; 

(2)  that  6  taught  ;>/; 

(3)  that  no  being  other  than  6  taught  jj'. 

This  analysis  in  which  the  describing  relation  is  Uac.imp- 
IS  typical  of  all  cases  in  which  a  descriptive  function  is 
used  in  a  proposition.   To  illustrate  a  mathematical  func- 
tion of  J)/,  for  ^eacAer-o/ substitute  ^reaier-Sj-^.^Aan-  so 
that  JI/-H3  stands  for  'i^e  quantity  that  is  greater  by  3 
than^.'  Again  for  the  predication  is-a-ScoUAman  substi- 
tute zs.dzvzstd/e.6y.4.   Thus,  in  place  of  the  proposition 
1  he  teacher  of  y  was  a  Scotchman,'  we  have  con- 
structed the  proposition  (6):  >  +  3  is  divisible  by  4,'  the 
full  force  of  which  is  rendered  as  follows : 

(^)    There  is  a  quantity,  say  6,  of  which  the  fol- 
lowmg  statements  may  be  made : 

'i)   that  (5  is  divisible  by  4; 

,2)   that  d  is  greater-by-3  than  y; 

(3)   that  no  quantity  other  than  ^j'is  preater-bv- v 
than^.  ^  ■^ 

5—2 


68 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


69 


Thirdly,  to  illustrate  a  prepositional  function,  for 
divisible-by-\  substitute  the  predicate  dubious ;  for  the 
quantitative  construct  >  +  3 '  substitute  the  prepositional 
construct  'y  is  p!  We  have  thus  constructed  the  se- 
condary proposition  (c)'.  'That  j  is/  is  dubious/  of 
which  the  full  force  is  rendered  as  follows : 

(c)  There  is  a  proposition,  say  b,  about  y,  of  which 
the  following  statements  may  be  made : 

( 1 )  that  b  is  dubious ; 

(2)  that  ^  predicates-/-aboutjj/; 

(3)  that  no  proposition  other  than  b  predicates- 
/-aboutj/. 

Now  in  example  {a)  the  ground  for  asserting  unique- 
ness of  the  construct  the  teacher  ofy  is  merely  empirical 
or  factual ;  but  in  example  {b)  the  necessary  and  sufficient 
condition  for  the  uniqueness  of  the  construct  jv  +  3  is  its 
mathematical  form,  as  indicated  by  the  symbol  + ;  and 
in  example  {c)  the  uniqueness  of  the  corresponding  con- 
struct y  is  p  similarly  depends  upon  Its  logical  form,  as 
indicated  by  the  logical  constant  is.    Dismissing  the  em- 
pirical example  which  requires  no  further  discussion,  it 
must  be  pointed  out  as  regards  the  quantitative  function 
{b)  and  the  prepositional  function  (c)  that  these  illustrate 
—not  quantitative  or  prepositional  functions  in  general 
—but  certain  specific  functions :  in  the  former  case  that 
which  is  constructed  by  means  of //^^,  and  in  the  latter 
case,  that  which  is  constructed  by  means  of  is.    The 
former  may  be  called  the  additive  and  the  latter  the 
characterising  function.    Just  as  the  quantitative  con- 
struct y-^-a  would  not  yield  a  quantity  unless  y  and  a 
were  themselves  quantities  of  the  same  kind ;  so  the 
prepositional  construct  y  is  p  would  not  yield  a  pro- 
position unless  the  two  constituents  y  and  /  were,  in 


¥ 


I 


their  nature,  relatively  to  one  another  as  substantive  to 
adjective.  A  specifically  different  form  of  quantitative 
construct  would  have  been  obtained  if  for  y-{-a  we  had 
substituted  y  :  a.  Similarly  a  specifically  different  pro- 
positional  form  of  construct  would  have  been  obtained 
if  for  s  is  p  we  had  substituted  x  is  identical  with  y. 
In  both  cases  the  uniqueness  of  the  construct  is  secured 
by  the  nature  of  the  operator  involved ;  viz.,  -h  which 
yields  a  sum,  or  :  which  yields  a  ratio  for  the  two  quanti- 
tative constructs  ;  and  is  and  is  identical  with  for  the 
two  prepositional  constructs.  If  there  is  any  difference 
between  the  uniqueness  of  the  prepositional  construct 
when  its  constituents  are  given  and  that  of  the  mathe- 
matical construct  when  its  constituents  are  given,  it  is 
that  the  uniqueness  in  the  former  case  is  assumed  en 
the  ground  of  its  intuitive  evidence  realised  in  the  mental 
act  of  constructing  the  proposition,  whereas  in  the  latter 
the  uniqueness  may  require  and  maybe  capable  of  formal 
demonstration. 

§  13.  Before  continuing  the  discussion  of  my  differ- 
ences from  Mr  Russell,  I  shall  examine  more  precisely 
what  he  means  by  a  descriptive  function.  A  descriptive 
function  (p.  245)  is  defined  to  be  a  phrase  of  the  kirni : 
'the  termor  that  has  the  relation  r  to  the  term  v.  In 
this  definition  the  sole  emphasis  is  to  be  laid  on  the 
predesignatien  the.  Now,  just  as  we  speak  of  the 
quantity  's-Vp^  so  we  speak  of  the  proposition  's  is  /.' 
But  these  quantitative  and  prepositional  phrases  differ 
from  ordinary  descriptive  phrases  such  as  the  writer 
of  Waver  ley'  or  'the  teacher  of  Xenophon '—which  are 
of  the  general  form  :  'the  thing  x  that  is  r  to  the  thing 
y' — in  that  they  do  not  explicitly  contain  any  descriptive 


70 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


71 


relation  r  (writing  or  teaching).    The  arithmetical  form 
's^p'  and  the  propositional  form   's  is  /'  having  in 
common  this  negative  characteristic,  I  shall  proceed  to 
maintain  that  they  are,  in  all  essential  logical  respects, 
identical  in  nature  ;  and,  if  either  of  the  two  can  be 
explicated  into  the  form  of  a  descriptive  function,  so 
can  the  other.    We  may  attempt  to  express  these  forms 
explicitly  as  descriptive  functions  by  introducing,  as 
the  describing  relation,  constructed  by.    Thus  the  pro- 
positional  function  maybe  rendered:  *the  proposition 
X  constructed  by  means  of  is  out  of  the  constituents  s 
and  p' ;  and  the  quantitative  function  may  be  rendered : 
*  the  quantity  x  constructed  by  means  oi plus  out  of  the 
constituents  s  and  /.'    This  attempt  reduces  the  state- 
ment of  equivalence  of  the  construct  with  the  proposed 
descriptive  phrase  to  a  mere  tautology  ;  for  'the  pro- 
position X  constructed  by  means  of  is  out  of  the  con- 
stituents s  and  p'  is  merely  a  lengthened  expression 
for  'the  proposition  s  is/';  and  similarly  'the  quantity 
X  constructed  by  means  of  plus  out  of  the  constituents 
s  and  / '  is  merely  a  lengthened  expression  for  '  the 
quantity  s-Vp,'     It   thus    turns   out   that   the   x  thus 
introduced    in   the  completely  formulated  descriptive 
phrase  stands  merely  for  the  function  itself,  i.e.  in  the 
one  case  for  's  is/'  and  in  the  other  for  ' s-\-p!   Follow- 
ing Russell  in  his  demand  that  a  descriptive  function 
must  only  be  defined  'in  use,'  the  statement  that  's  is 
p  is  dubious*  or  that  's-\-p  is  divisible  by  4'  must  be 
rendered  'the  proposition  x  constructed  etc.  (as  above) 
is  dubious,'  or  'the  quantity ^  constructed  etc.  (as  above) 
is  divisible  by  4.'    In  this  way  the  original  statements: 
'the  proposition  5  is  /  is  dubious,'  and  'the  quantity 


■'^^ 


s-\-p  is  divisible  by  4,'  which  were  supposed  to  require 
definition,  are  after  all  defined  tautologically. 

§  14.  Another  way  of  attacking  Russell's  proposi- 
tional function,  which  in  fact  presents  only  another 
aspect  of  the  same  criticism,  is  to  ask:  What  are  the 
variants  for  any  given  proposional  function,  and  what 
function  is  it  that  a  given  propositional  form  exhibits  t 
In  his  first  introduction  of  the  notion  of  propositional 
function,  Mr  Russell  gives  three  quite  different  appli- 
cations of  the  symbol  for  a  function.  According  to  his 
first  definition,  ^  is  called  a  propositional  function  when 
X  is  variable  provided  that  when  x  is  replaced  by  the 
constant  a,  ^a  represents  a  proposition.  Now  here  the 
symbol  for  a  function  is  first  used  along  with  a  variable 
and  then  along  with  a  constant;  although  Russell  insists 
that  ^a  is  not  a  function  but  a  proposition,  and  that  <^ 
is  not  a  proposition  but  a  function.  It  seems  to  me  that 
he  cannot  attach  the  symbol  for  a  function  exclusively 
to  a  variable  in  this  way  without  contradiction  at  every 
point;  and  it  is  for  this  reason  that,  in  my  account  of 
functions,  I  have  used  the  word  variant  to  include  both 
Russell's  variable  and  his  constant.  There  is  yet  a  third 
application  of  the  symbol  for  a  function  deliberately 
introduced  in  the  very  first  paragraph  of  his  exposition, 
by  way  of  correcting  his  initial  definition  of  propositional 
function.  For  his  first  account  is  that  ^x  is  to  be  called 
a  propositional  function,  owing  to  the  ambiguity — or  as 
I  should  prefer  to  say  indeterminateness — of  the  symbol 
X,  and  that  it  is  not  itself  a  proposition,  and  would  only 
become  a  proposition  when  a  is  substituted  for  x.  Ihis 
is  corrected,  however,  when  he  takes  the  example  'x 
is  hurt'  which  he  says  illustrates,  not  a  propositional 


72 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


73 


function,  but  an  ambiguous  (i.e.  an  indeterminate)  value 
of  a  propositional  function.  Thus,  as  I  have  pointed 
out,  he  illustrates  the  use  of  the  word  function  in  his 
first  paragraph  in  three  different  ways  which  are  sym- 
bolised as  follows :  'a  is  hurt,'  'x  is  hurt,'  and  '5t  is  hurt.' 
The  last  application  of  the  word  function  is  that  which 
he  wishes  to  be  finally  adopted;  but,  in  spite  of  this,  he 
continually  uses  the  word  in  both  of  the  two  other  ap- 
plications. It  is  still  more  surprising  that,  on  page  6  of 
his  Introduction,  where  he  gives  a  preliminary  account 
of  the  ideas  and  notations  of  logical  symbolism,  he  uses 
the  word  function  without  any  explanation  of  its  meaning, 
and  in  deliberate  defiance  of  his  own  later  definition. 
Thus  he  speaks  of  the  fundamental  functions  of  pro- 
positions in  these  words :  'an  aggregation  of  propositions 
considered  as  wholes,  not  necessarily  unambiguously 
determined,  into  a  single  proposition  more  complex  than 
its  constituents,  is  a  function  with  propositions  as  argu- 
ments.' This  account  appears  clearly  to  suggest  that  un- 
ambiguously determined  constituents  are  allowable  as 
arguments  for  a  function,  which  contradicts  his  explicit 
definition.  He  proceeds  to  enumerate  the  four  funda- 
mental functions  of  propositions  which  are  of  logical  im- 
portance, viz.  (i)  the  contradictory  function,  which  I 
have  called  the  negative  function ;  (2)  the  logical  sum 
or  disjunctive  function,  which  I  have  called  the  alter- 
native function;  (3)  the  logical  product,  which  both  he 
and  I  call  the  conjunctive  function ;  (4)  the  implicative 
function,  for  which  I  have  used  the  same  term.  These 
four  functions  I  have  called  conjunctional  functions,  in 
contrast  to  the  one  fundamental  predicational  function. 
The  recognition  of  this  distinction,  which  does  not  appear 


j«  .' 


% 


in.  Mr  Russell's  account,  would  have  simplified  and 
corrected  his  *  theory  of  types.'  But,  in  thus  introducing 
the  specific  conjunctional  functions,  he  inevitably  adopts 
the  familiar  meaning  of  the  mathematical  term  'function,' 
the  essence  of  which  lies,  not  in  the  indeterminateness 
of  the  constituent  terms,  but  in  the  identity  of  form  that 
is  exhibited  in  the  process  of  substituting  indifferently 
any  one  term  for  any  other. 

That  he  is  not  only  in  disagreement  with  universal 
usage,  but  also  logically  mistaken,  when  he  says  that 
it  is  a  function  of  which  the  essential  characteristic  is 
ambiguity — and  thus  that  <^  ambiguously  denotes  ^a, 
(f)dy  <l>Cy  where  <f>ay  (f>d,  (f>c  are  the  various  values  of  (f>x — 
is  shown  by  noting  that  the  ambiguity  attaching  to  (fxx 
is  not  due  to  the  nature  of  <^  as  a  function,  but  to  the 
nature  of  the  symbol  x  itself;  that  is  to  say,  <f>x  am- 
biguously denotes  <^^,  (f)d,  (f)Cy  etc.,  only  because  x  am- 
biguously denotes  a,  b,  c,  etc.  In  short  a  propositional 
function  has  ambiguous  denotation,  if  it  contains  a  term 
having  ambiguous  denotation ;  whereas  a  propositional 
function  has  unambiguous  denotation,  if  it  contains  no 
term  having  ambiguous  denotation. 

§  15.  Hitherto,  in  illustrating  Russell's  account,  we 
have  taken  the  propositional  function  to  be  a  function 
of  a  single  variable,  viz.,  of  the  symbol  for  the  subject 
of  the  proposition,  the  predicate  standing  for  a  constant 
It  is  obvious,  however,  that  no  proposition  can  be  re- 
»  garded  as  a  function  of  a  single  variant  unless  the  pro- 
position is  represented  by  a  simple  letter;  and  we  will 
therefore  take  the  specific  propositional  form  'x'xsp'  to 
illustrate  a  function  of  two  variables.  The  variants  of 
which  this  is  a  function  would  naturally  be  taken  as  the 


74 


CHAPTER  III 


SYMBOLISM  AND  FUNCTIONS 


75 


symbols  x  and  p  themselves ;  but,  since  Russell  refuses 
to  allow  a  predicate  or  adjective  to  stand  by  itself,  he 
takes  as  the  two  variables  the  subject  term  x  together 
with  the  symbolic  variable  *^  is  /.'  The  symbolic  ex- 
pression 'x\sf  may  be  read  '^r-blank  is  /';  by  which 
is  meant  that  instead  of  the  full  propositional  form  'x 
is  /,'  we  suppose  that  the  subject-term  x  is  omitted, 
leaving  a  blank.  But,  if  we  use  a  blank  symbol  for  the 
subject-term,  we  ought  in  consistency  to  be  allowed  to 
use  a  similar  blank  symbol  for  the  predicate  term.  This 
would  give  rise  to  nine  combinations  all  of  which  are  of 
the  same  propositional  form:  'this  is  hurt,'  'x  is  hurt,' 
'this  is/,'  '^is  hurt,'  'this  is/,'  'Jf  is/,'  'x  is/,'  'x  is/,' 
and  finally  'x  is/.'  Of  these  nine  phrases,  Russell  uses 
only  'this  is  hurt,'  'x  is  hurt'  and  'x  is  hurt';  of  which 
the  two  latter  illustrate  the  two  admittedly  different 
meanings  or  applications  of  the  general  notion  ^x,  i.e. 
of  the  propositional  function.  Now,  though  his  first 
reference  is  to  a  propositional  function  taking  a  single 
argument,  nevertheless  he  allows  that  any  proposition 
(as  distinguished  from  a  propositional  function)  when 
analysed  contains  at  least  two  constituents.  For  example, 
the  proposition  'this  is  hurt'  as  analysed  contains  the 
two  constituents  'this'  and  'x  is  hurt.'  In  my  view, 
there  is  no  ground  whatever  for  preferring  this  analysis 
either  to  that  in  which  the  constituents  are  'hurt'  and 
'this  is/,'  or  to  that  in  which  the  constituents  are  'this 
is/'  and  'jc  is  hurt.'  But,  returning  to  his  own  analysis 
in  which  'this'  and  'x  is  hurt'  are  assigned  as  the  two 
constituents  of  'this  is  hurt,'  as  also  'x'  and  '^  is/' 
as  the  two  constituents  of  '  x  is  / ' ;  we  must  insist 
upon  asking :  What  is  the  specific  function  for  the 


'W 


case  of  the  proposition  ';r  is/'  when  its  two  arguments 
are  taken  to  be  'x'  and  '&  is/'?  Mr  Russell  only  tells 
us  that  'jr  is/'  =  '/{x,  ^  is/)'  where  the  specific  symbol 
yhas  nowhere  been  defined  by  him.  It  is  as  if  he  had 
said  that  the  quantitative  function  'x-Vp'  has  for  its 
two  constituents,  variants  or  arguments:  (i)  ^  and  (2) 
x+/.  Now  according  to  this  analysis  of  the  nature  of 
a  function,  the  process  by  which  a  function  is  constructed 
out  of  two  variables  is  to  substitute  in  one  of  these 
variables  x  for  x^  so  that  taking  a  similar  example  to 
the  above,  the  constituents  of  the  quantitative  construct 
'x-r-p'  would  be  x  and  x-^p.  Every  mathematician 
would  take  as  the  two  constituents  of  the  construct 
x-=rp  the  two  simple  symbols  x  and/ ;  as  Russell  himself 
does  in  his  preliminary  account  of  the  alternative  func- 
tion X  or/,  of  which  the  two  constituents  are  the  simple 
symbols  x  and  /.  In  fact  he  can  only  take  a  function 
of  a  single  variable  as  ambiguously  denoting  a  pro- 
position, by  starting  with  what  I  have  called  a  non- 
formal  function,  e.g.  'x  is  hurt'  as  a  non-formal  function 
oix\  instead  of  starting  with  the  essentially  logical  notion 
of  a  function,  which  is  synonymous  with  the  form  of  a 
construct  such  as  'jr  is/'  where  instead  of  one  material 
or  variable  constituent  there  are  two.  In  short  the  form 
of  a  proposition,  if  it  has  form  at  all  and  is  not  simply 
expressed  by  a  simple  symbol,  must  contain  two  inde- 
pendent constituents.  When  Mr  Russell  says  that 
^  {x)  is  a  propositional  function,  provided  that  <^  [a)  is 
a  proposition,  he  provides  us  with  no  indication  as  to 
the  form  that  <^  {a)  must  assume  in  order  that  ^  {a) 
shall  constitute  a  proposition. 


r 


THE  CATEGORICAL  SYLLOGISM 


n 


CHAPTER  IV 

THE  CATEGORICAL  SYLLOGISM 

§  I.    As  the  relation  between  implication  and  in- 
ference has  already  been  explained,  we  may  treat  the 
syllogism  indifferently  as  a  species  either  of  implication 
or  of  inference:  regarded  as  implication,  the  propositions 
concerned  must  be  spoken  of  as  implicants  and  impli- 
cate; regarded  as  inference,  we  speak  of  them  as  pre- 
misses and  conclusion.   The  term  syllogism  is  strictly 
confined  to  one  only  of  the  many  forms  of  demonstrative 
inference;  and  in  this  strict  usage  must  be  defined  as 
an  argument  containing  two  premisses  and  a  conclusion, 
involving  between  them  three  terms,  each  of  which 
occurs  in  two  different  propositions.  That  occurring  as 
predicate  in  the  conclusion  is  called  the  major  term; 
that  occurring  as  subject  in  the  conclusion,  the  minor 
term;  and  that  not  occurring  in  the  conclusion,  the 
middle  term.    The  distinction  between  the  major  and 
minor  terms  determines  which  of  the  premisses  shall  be 
called  major  and  which  minor:    that  which  contains 
the  predicate  of  the  conclusion  being  called  the  major 
premiss;  and  that  which  contains  the  subject  of  the 
conclusion  being  called  the  minor  premiss.    Reference 
to  the  conclusion  is  thus  required  before  the  premisses 
can  be  distinguished  as  major  or  minor.   The  canonical 
order  of  the  three  propositions,   viz.   major  premiss, 
minor    premiss,    conclusion,    is   purely   artificial,    and 
adopted  only  for  general  purposes  of  reference.    The 
mood  of  a  syllogism  is  defined  by  the  forms  [A,  E,  /, 


^ 


#-  "J 


*•>, 


jor  O)  of  the  three  propositions  constituting  the  major 
premiss,  minor  premiss,  and  conclusion,  in  their  canoni- 
cal order.  Furthermore  syllogisms  are  distinguished 
according  to  figure :  the  first  figure  being  that  in  which 
the  middle  term  occurs  as  subject  in  the  major  premiss 
and  predicate  in  the  minor ;  the  fourth  figure  being  that 
in  which  the  middle  term  occurs  as  predicate  in  the 
major  and  as  subject  in  the  minor:  the  second  figure, 
that  in  which  the  middle  term  occurs  as  predicate  in 
both  premisses;  and  the  third  figure  that  in  which  the 
middle  term  occurs  as  subject  in  both  premisses.  Two 
syllogisms  would  be  said  to  be  of  different  form,  although 
they  might  agree  in  mood,  provided  they  differed  in 

figure. 

§  2.  There  are  two  opposite  tendencies  in  the  choice 
of  illustrations  of  the  syllogism,  both  of  which,  in  my 
view,  should  be  avoided.  The  first  is  to  select  examples 
composed  of  propositions,  each  of  which  is  universally 
accepted  as  true.  But  such  illustrations  hinder  the 
learner  from  examining  the  validity  of  the  inferential 
process  from  premisses  to  conclusion,  since  he  is  apt  to 
assume  validity  because  of  his  familiarity  with  the  pro- 
positions as  being  generally  accepted.  The  opposite 
course,  which  we  find  amusingly  illustrated  by  Lewis 
Carroll,  is  to  select  propositions  which  are  obviously 
false.  But  this  leads  the  learner  to  regard  the  syllogism 
merely  as  a  kind  of  game,  and  as  having  no  real  signi- 
ficance in  actual  thought  procedure.  It  is  preferable, 
therefore,  to  select  propositions  which  are  dubious, 
or  which  are  affirmed  by  some  persons  and  denied 
by  others.  Of  such  propositions  important  kinds  are 
(i)  those  which  deal  with  political,  ethical,  or  similar 


78 


CHAPTER  IV 


/ 


THE  CATEGORICAL  SYLLOGISM 


79 


topics  m  general,  e.g.  *  Lying  is  sometimesi  right,'  *A11 
countries  that  adopt  free-trade  are  prospe-rous,'  'The 
suffrage  should  not  be  extended  to  uneducated  persons; 
(2)  those  which  exercise  the  faculty  of  jud^rnent,  in  the 
Kantian  sense,  upon  some  individual  case^  e.g.  *This 
man  is  untrustworthy/  ^The  Niche  is  finen*  than  the 
Venus  of  Milo/  'Esau  is  a  more  lovable  per-son  than 
Jacob.'  y 

§  3.  Correlative  to  the  syllogism  we  may  heje  in- 
troduce the  antilogism,  in  reference  to  which  the  afoove 
principle  of  selecting  examples  will  be  seen  to  ha^^e 
special  significance.  An  antilogism  may  be  defined  as  a 
formal  disjunction  of  two,  three,  or  more  propositions, 
each  of  which  is  entertained  hypothetically.  When 
limited  to  three  propositions  constituting  a  disjunctive 
trio,  the  antilogism  may  be  formulated  in  terms  of  illus- 
trative symbols  as  follows:  *the  three  propositions/*,  Qy 
and  R  cannot  be  true  together.'  It  is  then  seen  that 
just  as  the  disjunction  oi  P  and  Q  is  equivalent  to  the 
implication  'If/*  is  true,  then  Q  is  false,'  so  the  disjunc- 
tion of  P,  Qy  and  R  is  equivalent  to  each  of  the  three 
implications : 

(i)    HP  and  Q  are  true,  then  R  is  false, 

(2)  HP  and  R  are  true,  then  Q  is  false, 

(3)  If  7?  and  Q  are  true,  then  P  is  false. 

We  may  put  forward  the  following  example  of  an 
antilogism,  no  one  of  the  propositions  of  which  would 
be  universally  acknowledged  either  as  true  or  as  false, 
but  which  taken  together  are  formally  incompatible : 

P.  All  tactful  persons  sometimes  lie. 
Q,   Lord  Grey  is  a  tactful  person. 
R,  Lord  Grey  never  lies. 


V 


w 


m-  ^ 


'%^ 


\\ 


* 

}* 


Something  could  be  said  in  support  of,  as  well  as  in 
opposition  to,  each  of  these  three  propositions;  but  it  is 
obvious  that  they  are  together  incompatible,  and  hence 
constitute  an  antilogism  or  disjunctive  trio.  This  anti- 
logism is  equivalent  to  each  of  the  three  following 
syllogistic  implications : 

1st      if  All  tactful  persons  sometimes  lie 
and  Lord  Grey  is  a  tactful  person, 
then  Lord  Grey  sometimes  lies. 

2nd    if  All  tactful  persons  sometimes  lie 
and  Lord  Grey  never  lies, 
then  Lord  Grey  is  not  a  tactful  person. 

3rd    if  Lord  Grey  never  lies 

and  Lord  Grey  is  a  tactful  person, 
then  Some  tactful  persons  never  lie. 

§  4.  The  propositions  in  each  of  these  syllogisms 
are  in  the  canonical  order  of  major,  minor,  conclusion, 
and  the  syllogisms  will  be  recognised  as  being  in  the 
first,  second,  and  third  figures  respectively.  In  defining 
the  figures  of  syllogism  we  may,  in  fact,  separate  the 
first  three  from  the  fourth  in  that  the  former  contain 
one  and  only  one  term  standing  in  one  proposition  as 
subject  and  in  another  as  predicate,  while  in  the  fourth 
figure  all  three  terms  occupy  this  double  position.  Such 
a  term  may  be  called  a  class-term,  on  the  ground  that 
a  class-term  has  a  partly  adjectival  meaning,  and  as  such 
serves  appropriately  as  predicate;  and  partly  a  sub- 
stantival meaning,  and  as  such  serves  appropriately  as 
subject.  The  first  three  figures,  then,  containing  only 
one  class-term,  are  distinguished  from  one  another 
according  as  this  term  occupies  one  or  another  position. 
In  the  first  figure  it  serves  as  the  middle  term;  in  the 


/ 


8o 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


8i 


second  figure  as  the  major  term ;  and  in  the  third  figure 
as  the  minor  term.  Taking  the  above  antilogism  as 
illustrative,  we  may  generalise  by  formulating  the 
following  antilogistic  dictum  for  the  first  three  figures : 

It  is  impossible  to  conjoin  together  the  three  pro- 
positions : 

Every  member  of  a  class  has  a  certain  property ; 
A  certain  object  is  included  in  that  class; 
This  object  has  not  that  property. 

From  this  single  antilogistic  dictum  we  construct  the 
dicta  for  the  first  three  figures  of  syllogism,  thus : 

Dictum  for  ist  Figure 

if  Every  member  of  a  class  has  a  certain  property 
and  A  certain  object  is  included  in  that  class, 
then  This  object  must  have  that  property. 

Dictum  for  2nd  Figure 

if  Every  member  of  a  class  has  a  certain  property 
and  A  certain  object  has  not  that  property, 
then  This  object  must  be  excluded  from  the  class. 

Dictum,  for  2^rd  Figure 

if  A  certain  object  has  not  a  certain  property 
and  This  object  is  included  in  a  certain  class, 
then  Not  every  member  of  the  class  has  that  property. 

These  dicta  bring  out  the  normal  function  of  each  of  the 
first  three  figures  in  thought-process.  Thus  we  are 
reasoning  in  the  first  figure  when,  having  established  a 
certain  characteristic  as  belonging  to  every  member  of 
a  class,  we  bring  forward  an  individual  object  known  to 
belong  to  the  class  and  proceed  to  assert  that  it  will 
have  the  characteristic  common  to  the  class.  We  are 
reasoning  in  the  second  figure  when,  having  similarly 


j^W 


established  a  certain  characteristic  as  belonging  to  every 
member  of  a  class,  and  having  found  that  an  individual 
object  has  not  this  characteristic,  we  proceed  to  assert 
that  it  does  not  belong  to  the  class.  We  are  reasoning 
in  the  third  figure  when  we  note  that  a  certain  object 
known  to  belong  to  a  certain  class  has  not  a  certain 
property,  and  proceed  to  assert  that  that  property  cannot 
be  predicated  universally  of  all  members  of  the  class ; 
or  otherwise,  when,  having  noted  that  an  object  known 
to  belong  to  a  certain  class  has  a  certain  character,  we 
infer  that  at  least  one  member  of  the  class  has  this 
character.  A  peculiarity  of  the  third  figure  is  that  it 
functions  either  destructively  or  constructively;  as  de- 
structive, it  disproves  some  universal  proposition  that 
may  have  been  suggested ;  as  constructive,  it  naturally 
suggests  the  replacement  of  the  particular  conclusion 
either  by  a  universal  whose  subject  is  restricted  by  some 
further  adjectival  characteristic,  or  by  an  unrestricted 
universal  to  be  obtained  by  induction  from  the  par- 
ticular conclusion. 

§  5.  A  second  illustration  of  an  antilogism  develop- 
ing into  three  syllogisms  may  be  chosen  with  the  purpose 
of  showing  how  purely  formal  and  elementary  reasoning 
underlies  even  the  most  abstract  arguments.    Thus : 

It  is  impossible  to  conjoin  the  three  propositions: 

P,    All  possible  objects  of  thought  are  such  as  have 
been  sensationally  impressed  upon  us; 

Q,    Substance  is  a  possible  object  of  thought ; 

R.    Substance  has  not  been  sensationally  impressed 
upon  us. 

Since  each  of  these  propositions  has  been  asserted  by 

J.  L.  H  6 


k 


82 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


83 


some  and  denied  by  other  philosophers,  the  three 
together  constitute  an  antilogism  having  the  same  illus- 
trative value  as  our  previous  example. 

Taking,  first,  P  and  Q  as  asserted  premisses  and 
nol-R  as  conclusion,  we  obtain  the  syllogistic  inference : 

P,    All  possible  objects  of  thought  have  been  sensa- 
tionally impressed  upon  us; 

Q.    Substance  is  a  possible  object  of  thought; 

.-.  not-^.    Substance  has  been  sensationally  impressed 
upon  us. 

With  some  explanations  and  modifications  this  syllo- 
gism represents  roughly  one  aspect  of  the  new  realistic 
philosophy. 

Taking,  next,  P  and  R  as  asserted  premisses  and 
not-^  as  conclusion,  we  have : 

P,    All  possible  objects  of  thought  have  been  sensa- 
tionally impressed  upon  us; 

R,    Substance  has  not  been  sensationally  impressed 
upon  us; 

.  •.  not-^.  Substance  is  not  a  possible  object  of  thought. 
This  syllogism  represents  very  fairly  the  position  of 
Hume. 

Taking,  lastly,  R  and  Q  as  asserted  premisses  and 
not-/^  as  conclusion,  we  have : 

R,    Substance  has  not  been  sensationally  impressed 
upon  us; 

Q,    Substance  is  a  possible  object  of  thought; 

. '.  not-/*.   Not  every  possible  object  of  thought  has  been 
sensationally  impressed  upon  us. 

This  syllogism  represents  almost  precisely  the  well- 
known  position  of  Kant. 


<\ 


^  -^ 


As  in  our  previous  example  these  three  syllogisms 
are  respectively  in  figures  i,  2,  and  3;  and,  moreover, 
Kant's  argument  in  figure  3  has  both  a  destructive 
function  in  upsetting  Hume*s  position;  and  a  construc- 
tive function  in  suggesting  the  replacement  of  the 
particular  conclusion  by  a  limited  universal  which  would 
assign  the  further  characteristic  required  for  discrimi- 
nating those  objects  of  thought  which  have  not  been 
obtained  by  experience  from  those  which  have  been 
thus  obtained. 

§  6.  Since  the  dicta,  as  formulated  above,  apply 
only  where  two  of  the  propositions  are  singular  or 
instantial,  they  must  be  reformulated  so  as  to  apply  also 
where  a//  the  propositions  are  general,  i.e.  universal  or 
particular.  Furthermore,  they  will  be  adapted  so  as  to 
determine  directly  all  the  possible  variations  for  each 
figure.    As  follows: 

Dictum  for  Fig,  i 

if  Every  one  of  a  certain  class  C  possesses  (or  lacks) 
a  certain  property  P 
and  Certain  objects  S  are  included  in  that  class  C, 
then  These  objects  5  must  possess  (or  lack)  that  pro- 
perty P. 

Dictum  for  Fig.  2 

if  Everyone  of  a  certain  class  C  possesses  (or  lacks) 
a  certain  property  P 
and  Certain  objects  5"  lack  (or  possess) that  property/^, 
then  These   objects   5*  must    be   excluded  from    the 
class  C 

Dictum  for  Fig,  3 

if  Certain  objects  5"  possess  (or  lack)  a  certain  pro- 
perty P 

6—2 


( 


84 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


85 


1 


\ 


A 


and  These  objects  kS  are  included  in  a  certain  class  C 
then  Not  every  one  of  the  class  C  lacks  (or  possesses) 
that  property  P. 
I.e.  Some  of  the  class  C  possess  (or  lack)  that  pro- 
perty P, 

In  each  of  these  dicta  the  word  'objects,'  symbolised 
as  5,  represents  the  term  that  stands  as  subject  in  both 
its  occurrences;  the  word  *  property'  P,  the  term  that 
stands  as  predicate  in  both  its  occurrences;  and  the 
word  'class'  C,  that  term  which  occurs  once  as  subject 
and  again  as  predicate.  Hence,  using  the  symbols 
5,  C,  P,  the  first  three  figures  are  thus  schematised : 


Fig.  I 
C-P 
S--C 

S-P 


Fig.  1 
C-P 
S-P 

s-c 


Fig.  3 
S-P 

S-C 
C-P 


§  7.  In  order  systematically  to  establish  the  moods 
which  are  valid  in  accordance  with  the  above  dicta,  it 
should  be  noted  in  each  figure  (i)  that  the  proposition 
S  —  P  is  unrestricted  as  regards  both  jguaHty  and 
quantity;  (2)  that  the  proposition  S—C  is  indepen- 
dently fixed  in  quality,  but  determined  in  quantity  by 
the  quantity  of  the  unrestricted  proposition ;  and  (3)  that 
the  proposition  C—  /*  is  independently  fixed  in  quantity, 
but  determined  in  quality  by  the  quality  of  the  un- 

.  restricted  proposition.  Thus  in  Fig.  i,  while  the 
conclusion  is  unrestricted,  the  minor  premiss  is  indepen- 

\  dently  fixed  in  quality  but  determined  in  quantity  by 
the  quantity  of  the  conclusion;  and  the  major  premiss 
is  independently  fixed  in  quantity  but  determined  in 

/quality  by  the  quality  of  the  conclusion.  In  Fig.  2, 
while  the  minor  premiss  is  unrestricted,  the  conclusion 


*  4 
I 


^11 


is  independently  fixed  in  quality  but  determined  in 
quantity  by  the  quantity  of  the  minor  premiss;  and  the 
major  premiss  is  independently  fixed  in  quantity,  but 
determined  in  quality  by  the  quality  of  the  minor  pre- 
miss. In  Fig.  3,  while  the  major  premiss  is  unrestricted, 
the  minor  premiss  is  independently  fixed  in  quality  but 
determined  in  quantity  by  the  quantity  of  the  major 
premiss,  and  the  conclusion  is  independently  fixed  in 
quantity  but  determined  in  quality  by  the  quality  of  the 

major  premiss. 

Having  in  the  above  dicta  italicised  the  phrase  in 
each  case  which  is  directly  restrictive,  the  proposition 
which  is  unrestricted,  i.e.  may  be  of  the  form  A  ox  E 
or  /or  O,  is  seen  to  be:  in  Fig.  i,  the  conclusion;  in 
Fig.  2,  the  minor  premiss ;  in  Fig.  3,  the  major  premiss. 
Hence  each  of  these  figures  contains  four  fundamental 
moods  derived  respectively  by  giving  to  the  unrestricted 
proposition  the  form  A,  E,  I  or  O,  Besides  these  four 
fundamental  moods  there  are  also  supernumerary  moods. 
These  are  obtained  by  substituting,  in  the  conclusion, 
a  particular  for  a  universal ;  or,  in  the  minor  premiss, 
a  universal  for  a  particular;  or,  in  the  major  again,  a 
universal  for  a  particular.  These  supernumerary  moods 
will  be  said  respectively  to  contain  a  weakened  con- 
clusion, a  strengthened  minor,  or  a  strengthened 
major;  and,  in  the  scheme  given  in  the  next  section, 
the  propositions  thus  weakened  or  strengthened  will 
be  indicated  by  the  raised  letters  w  or  ^  as  the  case 

may  be. 

§  8.  Adopting  the  method  above  explained,  we  may 
now  formulate  the  special  rules  for  determining  the 
valid  moods  in  each  figure  as  follows : 


86 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


87 


Rules  for  Fig,  i. 

The  conclusion  being  unrestricted  in  regard  both  to 
quality  and  quantity, 

{a)  The  major  premiss  must  in  quantity  be  uni- 
versal, and  in  quality  agree  with  the  conclusion. 

{b)  The  minor  premiss  must  be  in  quality  affirma- 
tive, and  in  quantity  as  wide  as  the  conclusion. 

Rules  for  Fig.  2. 

The  minor  premiss  being  unrestricted  in  regard  both 
to  quality  and  quantity, 

{a)  The  major  premiss  must  be  in  quantity  uni- 
versal, and  in  quality  opposed  to  the  minor. 

(^)  The  conclusion  must  be  in  quality  negative, 
and  in  quantity  as  narrow  as  the  minor. 

Rules  for  Fig.  3. 

The  major  premiss  being  unrestricted  in  regard  both 
to  quantity  and  quality, 

(a)  The  conclusion  must  in  quantity  be  particular, 
and  in  quality  agree  with  the  major, 

(Ji)  The  minor  premiss  must  in  quality  be  affirma- 
tive, and  in  quantity  overlaps  the  major. 

Italicising  in  each  case  the  unrestricted  proposition, 
we  may  represent  the  valid  moods  for  the  first  three 
figures  in  the  following  table: 

Valid  Moods  for  the  "  One- Class  "  Figures. 


Fundamentals 

Supernumeraries 

Fig.  I 

KkA 

EA^" 

AI/ 

ElO 

SW 

AAI 

sw 
EAO 

Fig.  2 

E/4E 

A^E 

E/0 

AOO 

sw 
EAO 

sw 
AEO 

Fig.  3 

All 

i5:io 

/AI 

OAO 

ss 
AAI 

ss 
EAO 

^  The  minor  and  major  will  necessarily  overlap  if  one  or  the  other 
is  universal,  not  otherwise. 


< 


§  9.  Having  established  the  valid  moods  of  the  first 
three  figures  from  a  single  antilogism,  we  proceed  to 
construct  those  of  the  fourth  figure  also  from  a  single 
antilogism;  thus: 

Taking  any  three  classes,  it  is  impossible  that 
The  first  should  be  wholly  included  in  the  second 
while  The  second  is  wholly  excluded  from  the  third 
and  The  third  is  partly  included  in  the  first. 

The  validity  of  this  antilogism  is  most  naturally 
realised  by  representing  classes  as  closed  figures.  Such 
a  representation  is  in  fact  valid,  although  the  relation 
of  inclusion  and  exclusion  of  classes  is  not  identical 
with  the  logical  relations  expressed  in  affirmative  and 
negative  propositions  respectively;  for,  there  is  a  true 
analogy  between  the  relations  between  classes  and  the 
relations  between  closed  figures;  in  that  the  relations 
between  the  relations  of  classes  are  identical  with  the 
corresponding  relations  between  the  relations  of  closed 
figures.  Thus  adopting  as  the  scheme  of  the  fourth 
figure : 

Ci  —  C3  Cg      C3  C3      Cj 

the  above  antilogism  will  be  thus  symbolised : 

It  is  impossible  to  conjoin  the  following  three  pro- 
positions : 

P.    Every  C^  is  C,, 

Q.   No  Q  is  C3, 
R.  Some  C^  is  Q. 
This  yields  the  three  fundamental  syllogisms 

(i)  If /^and  Qy  then  not-R]  i.e. 

if  Every  Q  is  C,  g  Q 

and  No  Q  is  Q,  p^^ 

then  No  Q  is  Cj.  ^  ^ 


88 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


89 


(2)  If  Q  and  R,  then  not-/*;  i.e. 

if  No  C2  is  C3 
and  Some  Cg  is  Ci, 
then  Not  every  C^  is  C. 

(3)  If/?  and  P,  then  not-^;  i.e. 

if  Some  C^  is  C^ 
and  Every  C^  is  Co, 
then  Some  C^  is  C3. 

Since  the  propositions  of  these  syllogisms  are 
arranged  in  canonical  order,  the  valid  moods  in  the 
fourth  figure  can  be  at  once  written  down :  AEE,  EIO, 
lAL  Moreover,  since  the  conclusion  of  the  first  mood 
is  universal,  it  may  be  weakened ;  since  the  minor  of 
the  second  is  particular,  it  may  be  strengthened ;  and 
since  the  major  of  the  third  is  particular,  it  also  may  be 
strengthened.    This  yields: 

Valid  Moods  of  the  Fourth  Figure. 

Fundamentals 


AEE        EIO 


lAI 


Supernumeraries 
w  s  s 

AEG        EAO        AAI 


Here  each  supernumerary  can  only  be  interpreted  in 
one  sense,  viz.,  as  containing  respectively  a  weakened 
conclusion,  a  strengthened  minor,  and  a  strengthened 
major.  In  contrast  to  this,  the  supernumeraries  of  the 
first  and  second  figures  must  be  interpreted  as  contain- 
ing either  a  weakened  conclusion  or  a,  strengthened 
minor;  and  those  of  the  third  figure  as  containing 
either  a  strengthened  major  or  a  strengthened  minor. 

§  10.  An  antiquated  prejudice  has  long  existed 
against  the  inclusion  of  the  fourth  figure  in  logical 
doctrine,  and  in  support  of  this  view  the  ground  that 
has  been  most  frequently  urged  is  as  follows: 


*• 


f 


W^, 


,.  ♦■ 


Any  argument  worthy  of  logical  recognition  must 
be  such  as  would  occur  in  ordinary  discourse.  Now  it 
will  be  found  that  no  argument  occurring  in  ordinary 
discourse  is  in  the  fourth  figure.  Hence,  no  argument 
in  the  fourth  figure  is  worthy  of  logical  recognition. 

This  argument,  being  in  the  fourth  figure,  refutes 
itself ;  and  therefore  needs  to  be  no  further  discussed. 

§  1 1.  Having  formulated  certain  intuitively  evident 
dicta,  the  observance  of  which  secures  the  validity  of 
the  syllogisms  established  by  their  means,  we  will  pro- 
ceed to  formulate  equally  intuitive  rules  the  violation 
of  which  will  render  syllogisms  invalid.  These  rules 
will  be  found  to  rest  upon  a  single  fundamental  con- 
sideration, viz.  if  our  data  or  premisses  refer  to  some 
only  of  a  class,  no  conclusion  can  be  validly  drawn 
which  refers  to  all  members  of  that  class.  This  is 
technically  expressed  in  the  rule: 

(i)  'No  term  which  is  undistributed  in  its  premiss 
may  be  distributed  in  the  conclusion.' 

This  rule  alone  is  not  sufficient  directly  to  secure 
validity,  but  from  it  we  can  deduce  other  directly 
applicable  rules  which,  taken  in  conjunction  with  the 
first,  will  be  sufficient  to  establish  directly  the  invalidity 
of  any  invalid  form  of  syllogism.  In  the  course  of 
deducing  these  other  rules  we  shall  make  use  of  certain 
logical  intuitions  that  are  obvious  apart  from  their  em- 
ployment in  this  deductive  process,  of  which  the  follow- 
ing may  be  mentioned: 

(a)  that  if  a  term  is  distributed  in  any  given 
proposition,  it  will  be  undistributed  in  the  contradictory 
proposition ;  and  conversely,  if  a  term  is  undistributed 
in  a  given  proposition,  it  will  be  distributed  in  the 


90 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


91 


contradictory  proposition.  That  this  is  so  is  directly  seen 
when  it  has  been  accepted  on  intuitive  grounds  that  only 
universals  distribute  the  subject  term,  and  only  nega- 
tives the  predicate  term ;  and  that  an  A  proposition  is 
contradicted  by  an  O,  and  an  /  proposition  by  an  E, 

(6)  That  any  syllogism  can  be  expressed  as  an 
antilogism  and  conversely.  This  principle  follows  from 
the  intuitive  apprehension  of  the  relation  between  im- 
plication and  disjunction. 

(c)  That  it  is  formally  possible  for  any  three 
terms  to  coincide  in  extension.  (This  particular  in- 
tuition is  employed  in  the  rejection  of  only  one  form  of 
syllogism.) 

We  are  now  in  a  position  to  deduce  from  our 
original  principle,  i.e.  from  rule  (i),  by  means  of  (a), 
{6)y  and  (^),  other  rules,  the  direct  application  of  which 
will  exclude  any  invalid  forms  of  syllogism. 

(2)  *Th^  middle  term  must  be  distributed  in  one  or 
other  of  the  premisses.* 

To  establish  this,  let  us  consider  the  antilogism 
which  disjoins  P,  Q  and  R\  this,  by  {b)  is  equivalent 
to  the  syllogism  'UP  and  Q,  then  not-R'  and  also  to 
the  syllogism  *If  P  and  R,  then  not-^.'  Taking  the 
first  of  these,  if  a  term  X  is  undistributed  in  the  premiss 
Py  it  must  be  undistributed  in  the  conclusion  not-^, 
i.e.  it  must,  by  {a),  be  distributed  in  R,  Applying  this 
result  to  the  second  syllogism  '  If  P  and  R,  then  not-QJ 
we  have  shown  that  if  the  middle  term  X  is  undistri- 
buted in  the  premiss  Py  it  must  be  distributed  in  the 
premiss  R,    This  then  establishes  rule  (2). 

(3)  *If  both  premisses  are  negative,  no  conclusion 
can  be  syllogistically  inferred.' 


W-' 


I 


For,  taking  any  two  universal  negative  premisses, 
these  can  be  converted  (if  necessary)  into  'No  P  is  M' 
and  *  No  5  is  J/* ;  which,  by  obversion,  are  respectively 
equivalent  to  'All  P  is  non-J/'  and  'All  6*  is  non-M' 
in  which  the  new  middle  term  non-J/  is  undistributed 
in  both  premisses.  But  this  breaks  rule  (2).  What 
holds  of  two  universals  will  hold  a  fortioH  if  one  or 
other  of  the  two  negative  premisses  is  particular.  Thus 
rule  (3)  is  established. 

(4)  *A  negative  premiss  requires  a  negative  con- 
clusion.* 

For,  taking  again  the  antilogism  which  disjoins  P, 
Q  and  R,  this  is  equivalent  both  to  the  syllogism  'UP 
and  Ry  then  not-^,'  and  to  the  syllogism  *If  P  and  Q, 
then  not-/?.'  Taking  the  first  of  these  two  syllogisms, 
by  rule  (3),  if  the  premiss  P  is  negative,  the  premiss  R 
must  be  affirmative.  Applying  this  result  to  the  second 
syllogism,  we  have,  if  the  premiss  P  is  negative,  the 
conclusion  not-/?  must  be  negative.  This  establishes 
rule  (4). 

(5)  'A  negative  conclusion  requires  a  negative 
premiss.* 

This  is  equivalent  to  the  statement  that  two  affirma- 
tive premisses  cannot  yield  a  negative  conclusion.  To 
establish  this  rule,  we  must  take  the  several  different 
figures  of  syllogism : 


Fig.  I 
M-P 
S-M 

S-P 


Fig.  2 
P-M 
S-M 

S-P 


Fig- 3 
M-P 

M-S 
S-P 


Fig.  4 
P-M 

M-S 
S-P 


For  the  first  or  third  figure,  affirmative  premisses 
with  negative  conclusion  would  entail  false  distribution 


92 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


93 


of  the  major  term ;  which  has  been  forbidden  under  our 
fundamental  rule  (i).  Taking  next  the  second  figure, 
it  would  entail  false  distribution  of  the  middle  term, 
forbidden  by  rule  (2).  Finally  taking  the  fourth  figure, 
it  would  either  entail  some  false  distribution  forbidden 
by  rules  ( i )  and  (2) ;  or  else  yield  the  mood  A  A  O  which 
would  constitute  a  denial  that  three  terms  could  coincide 
in  extension,  thus  contravening  (^).    This  establishes 

rule  (5). 

§  12.  The  five  rules  thus  established  may  be  re- 
arranged and  summed  up  into  two  rules  of  quality  and 
two  rules  of  distribution,  viz. 

A.  Rules  of  Quality, 

(a^  For  an  afifirmative  conclusion  both  premisses 
must  be  affirmative. 

(^2)  For  a  negative  conclusion  the  two  premisses 
must  be  opposed  in  quality. 

B.  Rules  of  Distribution. 

(6^)  The  middle  term  must  be  distributed  in  at 
least  one  of  the  premisses. 

(4)  No  term  undistributed  in  its  premiss  may  be 
distributed  in  the  conclusion. 

These  rules  having  been  framed  with  the  purpose  of 
rejecting  invalid  syllogisms,  we  may  first  point  out  that, 
irrespective  of  validity,  there  are  sixty-four  abstractly 
possible  combinations  of  major,  minor  and  conclusion. 
The  Rules  of  Quality  enable  us  to  reject  en  bloc  all 
moods  except  those  coming  under  the  following  three 
heads,  viz.  those  which  contain  (i)  an  affirmative  con- 
elusion  (requiring  affirmative  major  and  affirmative 
minor) ;    (ii)    a  negative  major  (requiring  affirmative 


^f 


1 


r 


^ 


\ 


minor  and  negative  conclusion);  (iii)  a  negative  m.inor 
(requiring  affirmative  major  and  negative  conclusion). 
This  leads  to  the  following  table,  which  exhibits  the 
24  possibly  valid  moods  unrejected  by  the  Rules  of 
Quality. 


Maj.  &  Min. 
Univ. 

Maj. 
Univ. 

Min. 
Part. 

Maj. 
Part. 

Min.    j 
Univ. 

Maj.  &  Min. 
Part. 

Concl.  Aflf. 

AAI 

AAA 

All 

AIA 

lAI 

lAA 

Ill 

IIA 

Maj.  Neg. 

EAO 

EAE 

EIO 

EIE 

GAG 

GAE 

GIG 

OIE 

Min.  Neg. 

AEO 

AEE 

AGO 

AOE 

lEG 

lEE 

IGG 

IGE 

§  1 3.  The  Rules  of  Quality  having  thus  been  applied, 
it  remains  to  reject  such  of  the  above  24  as  violate  the 
Rules  of  Distribution,  (i)  for  the  middle  term,  (ii)  for 
the  major  term,  (iii)  for  the  minor  term.  This  will  re- 
quire three  special  rules  for  each  of  the  four  figures: 

Fig-  3 
M-P 

M-s 
s-P 


Fig.  I 

M-P 

S-M 

S-P 


Fig.  2 

P-M 

S-M 


S-P 


Fig- 4 
P-M 

M-S 
S-P 


The  above  scheme  shows  that  it  will  be  convenient 
to  bracket  Fig.  i  with  Fig.  4  for  the  middle  term,  Fig.  i 
with  Fig.  3  for  the  major  term,  and  Fig.  i  with  Fig.  2 
for  the  m,inor  term\  leading  to  the  following: 

Rules  for  correct  Distribution. 
\st  of  the  Middle  Term, 

Fig.  I.  If  the  minor  is  affirmative,  the  major  must 
be  universal. 

Fig.  4.  If  the  major  is  affirmative,  the  minor  must 
be  universal. 


:/ 


94 


CHAPTER  IV 


Fig.  2.  One  premiss  must  be  negative;  i.e.  con- 
clusion must  be  negative. 

Fig.  3.  One  or  the  other  of  the  premisses  must  be 
^universal. 

2nd  of  the  Major  Term, 

Figs.  I  and  3.  If  the  conclusion  is  negative,  the 
major  must  be  negative;  i.e.  (in  either  case)  the  minor 
must  be  affirmative. 

Figs.  2  and  4.  If  the  conclusion  is  negative,  the 
major  must  be  universal. 

'i^rd  of  the  Minor  Term. 

Figs.  I  and  2.  If  the  minor  is  particular,  the  con- 
clusion must  be  particular. 

Figs.  3  and  4.  If  the  minor  is  affirmative,  the  con- 
clusion must  be  particular. 

These  rules  have  been  grouped  by  reference  to  the 
term  (middle,  major  or  minor)  which  has  to  be  correctly 
distributed.  They  will  now  be  grouped  by  reference  to 
the  figure  (ist,  2nd,  3rd  or  4th)  to  which  each  applies. 
In  this  rearrangement  we  shall  also  simplify  the  for- 
mulations by  replacing  where  possible  a  hypothetically 
formulated  rule  by  one  categorically  formulated.  As  a 
basis  of  this  reformulation  we  take  the  rules  of  quality 
for  Figs.  I,  2  and  3,  which  have  already  been  expressed 
categorically;  viz.  for  Figs,  i  and  3:  'The  minor  pre- 
miss must  be  affirmative,'  and  for  Fig.  2:  *The  con- 
clusion must  be  negative.'  Conjoining  the  categorical 
rule  (of  quality)  for  Fig.  i  with  its  hypothetical  rule, 
'  If  the  minor  is  affirmative  the  major  must  be  universal,' 
we  deducefor  thisfigure  the  categorical  rule  (of  quantity), 
'The  major  must  be  universal'    Again,  conjoining  the 


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96 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


97 


categorical  rule  (of  quality)  for  Fig.  2  with  its  hypo- 
thetical rule  *If  the  conclusion  is  negative  the  major 
must  be  universal,'  we  deduce  for  this  figure  the  cate- 
gorical rule  (of  quantity),  *The  major  must  be  universal/ 
Lastly,  conjoining  the  categorical  rule  (of  quality)  for 
Fig.  3  with  its  hypothetical  rule,  4f  the  minor  is  affir- 
mative the  conclusion  must  be  particular,'  we  deduce 
the  categorical  rule  (of  quantity)  for  this  figure,  *The 
conclusion  must  be  particular.'  The  remaining  rules 
must  be  repeated  without  modification. 

The  Special  Rules  of  Distribution  for  each  Figure 
and  the  application  of  these  rules  of  distribution  to  the 
scheme  of  possibly  valid  moods  unrejected  by  the  rules 
of  quality  are  set  out  on  the  preceding  page. 

§  14.  We  will  now  compare  the  results  reached  by 
the  two  methods — direct  and  indirect.  The  direct 
method  determines,  by  means  of  certain  intuited  dicta, 
what  moods  are  to  be  accepted  as  valid ;  the  indirect 
method  determines — on  equally  intuitive  principles — 
what  moods  are  to  be  rejected  as  invalid,  and  conse- 
quently what  moods  remain  unrejected.  We  gather  from 
this  comparison  that  the  24  moods  (6  for  each  figure) 
that  are  established  as  valid  by  the  direct  method  are 
identical  with  the  24  that  are  not  rejected  as  invalid  by 
the  indirect  method.  It  follows  that  the  two  methods 
must  be  used  as  supplementary  to  one  another.  For, 
apart  from  the  use  of  the  indirect  method  we  should 
not  have  proved  that  the  moods  established  as  valid 
were  the  only  valid  moods ;  and  apart  from  the  use  of 
the  direct  method  we  should  not  have  proved  that  the 
moods  unrejected  as  invalid  were  themselves  valid.  In 
short,  by  the  direct  method  we  establish  the  conditions 


r 


that  are  sufficient  to  ensure  validity,  and  by  the  indirect 
those  that  are  necessary  to  ensure  validity. 

§  1 5.  The  attached  diagram,  taking  the  place  of  the 
mnemonic  verses,  indicates  which  moods  are  valid,  and 
which  are  common  to  different  figures.  The  squares  are 
so  arranged  that  the  rules  for  the  first,  second  and  third 
figures  also  show  the  compartments  into  which  each 


4TH 


mood  is  to  be  placed,  according  as  its  major,  minor  or 
conclusion  is  universal  or  particular,  affirmative  or  nega- 
tive. The  valid  moods  of  the  fourth  figure  occupy  the 
central  horizontal  line. 

§  16.  A  very  simple  extension  of  the  syllogism  and 
of  the  corresponding  antilogism  is  treated  in  ordinary 
logic  under  the  name  Sorites,  which  is  a  form  of  argu- 
ment comprised  of  propositions  forming  a  closed  chain ; 

J.  L.  n  7 


98 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


99 


and  may  be  defined  as  'an  argument  containing  any 
number  of  terms  and  an  equal  number  of  propositions, 
such  that  each  term  occurs  twice  and  is  linked  in  one 
proposition  with  one  term,  and  in  another  with  a  different 
term.'  E.g.  an  argument  of  this  form,  containing  five 
terms,  would  be  represented  by  the  five  propositions: 
a — b,  b — c,  c — dy  d—e,  e — a,  where  each  term  placed 
first  may  stand  indifferently  either  for  subject  or  for 
predicate.  Now  it  will  be  found  that  the  necessary  and 
sufficient  rules  for  inferences  of  this  form  are  virtually 
the  same  as  for  the  three-termed  argument;  viz. 

A.  Rules  of  Quality, 

(^i)  For  an  affirmative  conclusion,  all  the  pre- 
misses must  be  affirmative. 

{a^  For  a  negative  conclusion,  all  but  one  of  the 
premisses  must  be  affirmative. 

B.  Rules  of  Distribution. 

(b^  Any  term  recurring  in  the  premisses  must  be 
distributed  in  (at  least)  one  of  its  occurrences. 

(^2)  Any  term  occurring  in  the  conclusion  must 
be  undistributed,  if  it  was  undistributed  in  its 
premiss. 

The  rules  for  the  corresponding  antilogism  reduce  to 

two,  viz. 

A.  Rule  of  Quality :  All  but  one  of  the  propositions 
must  be  affirmative. 

B.  Rule  of  Distribution:  Every  term  must  be  dis- 
tributed at  least  once  in  its  two  occurrences. 

§  17.  There  are  certain  irregular  forms  of  syllogism 
or  of  sorites,  which  may  be  reduced  to  strict  syllogistic 
form  by  the  employment  of  certain  logical  principles, 
the  nature  of  which  we  shall  proceed  to  discuss.    The 


i 


J 


f 


1 


arguments  to  be  considered  are  those  which  involve  a 
larger  number  of  terms  than  of  propositions;  and  it  is 
necessary,  in  order  to  test  the  validity  or  invalidity  of 
such  arguments,  to  substitute  if  possible,  for  one  or 
more  of  the  propositions,  an  equivalent  proposition, 
which  will  diminish  the  terms  to  the  number  of  pro- 
positions. This  is  done  by  means  of  obversion,  con- 
version, and  other  logical  modes.  Until  this  substitution 
is  made,  the  argument  may  be  valid,  and  yet  break  one 
or  more  of  the  rules  of  syllogism.  Thus  two  of  the 
premisses  may  be  negative,  and  the  argument  yet  be 
valid,  the  apparent  violation  of  the  rule  being  due  to 
the  presence  of  more  than  the  proper  number  of  terms ; 
for  example, 

No  right  action  is  inexpedient, 
This  is  not  a  wrong  action, 
•  '.  This  is  expedient. 

Here  by  merely  ob verting  the  two  premisses  we  arrive 
at  the  standard  syllogism  of  the  first  figure,  namely : 

Every  right  action  is  expedient. 
This  is  a  right  action, 
.'.  This  is  expedient. 

In  all  cases  of  substituting  for  a  proposition  some  equi- 
valent, we  may  require,  besides  simple  conversion,  the 
replacement  of  some  term  by  one  of  its  cognates.  Thus 
in  obversion,  we  replace  P  by  nol-P  or  conversely ;  P 
and  xiQ\.-P  being  the  simplest  case  of  cognates.  Again 
any  relative  term  may  be  replaced  by  its  cognate  co- 
relative.  Now  in  the  previous  illustration  obversion 
alone  was  required,  whereas  if  the  major  had  btt;!i 
written'  No  inexpedient  action  is  right'  conversion  would 
have  been  required  before  obverting.    To  illustrate  the 

7—2 


100 


CHAPTER  IV 


THE  CATEGORICAL  SYLLOGISM 


lOI 


replacement  of  a  relative  by  its  corelative,  we  may  take 
the  old  example  from  the  Port  Royal  logic, 

The  Persians  worship  the  sun, 
The  sun  is  a  thing  insensible, 
.-.  The  Persians  worship  a  thing  insensible. 

This  argument  contains  five  terms,  viz.,  the  Persians, 
worshippers  of  the  sun,  the  sun,  a  thing  insensible,  and 
worshippers  of  a  thing  insensible.  The  process  of  re- 
ducing this  argument  to  a  strictly  three-termed  argument 
is  effected  by  what  is  called  ^relatively  converting'  the 
major  premiss,  and  again  *  relatively  converting'  the 
conclusion  syllogistically  arrived  at  from  our  new  pre- 
misses. The  transformed  argument  then  assumes  the 
form  of  a  strict  syllogism  in  the  third  figure : 

The  sun  is  worshipped  by  the  Persians, 

The  sun  is  a  thing  insensible, 
.-.A  thing  insensible  is  worshipped  by  the  Persians, 

where,  by  converting  the  conclusion,  we  reach  that  re- 

'  The  Persians  worship  a  thing  insensible. 
§  1 8.  The  question  whether  the  syllogism  is  actually 
used  in  thought  process  is  met  by  noting  that,  while  in 
ordinary  discourse  it  is  rare  to  find  three  propositions 
constituting  a  syllogism  explicitly  propounded,  argu- 
ments of  a  syllogistic  nature  are  of  frequent  occurrence. 
These  syllogisms  are  expressed  as  enthymemes,  i.e  with 
the  omission  of  one  at  least  of  the  requisite  propositions.  / 
Now  in  an  enthymeme  there  is  one,  and  only  one,  pro- 
position which  could  be  introduced  to  render  the  corre- 
sponding syllogism  valid.  For  this  reason  the  enthy- 
meme  is  liable  to  one  or  other  of  two  forms  of  attack : 
first  it  may  be  attacked  on  the  ground  that  the  premiss 


li 


V 


K 


1 


supplied  by  the  hearer  is  true,  and  yet  renders  the  argu- 
ment invalid ;  or  secondly,  that  the  premiss  supplied  by 
the  hearer  is  false,  and  is  yet  the  onlv  one  which  would 
render  the  argument  valid.  The  former  case  w(3uld  be 
said  to  involve  a  formal  fallacy;  and  the  latter  a  matericil 
fallacy.  For  example:  Consider  the  enthymeme,  1  his 
flower  is  a  labiate,  because  it  is  square-stalked.  Here 
the  premiss  'All  labiates  are  square-stalked,'  which  is 
true,  renders  the  argument  formally  invalid  ;  on  the 
other  hand,  the  proposition  which  renders  the  argument 
formally  valid,  namely  'All  square-stalked  plants  are 
labiates'  is  false.  These  fallacies  arise,  for  the  most  part, 
in  the  case  of  disagreement  between  disputants  with 
respect  to  the  conclusion.  An  enthymeme  is  free  from 
both  kinds  of  fallacy  when  the  premiss  to  be  supplied 
is  known  or  accepted  by  all  parties,  and  at  the  same 
time,  renders  the  argument  formally  valid.  Thus  a 
strictly  valid  argument  is  expressed  in  the  form  of  an 
enthymeme  when  there  is  no  question  with  regard  to 
the  truth  of  the  omitted  proposition  which  will  render 
the  argument  formally  valid. 

This  may  be  instructively  illustrated  by  taking  ex- 
amples where  either  the  major  or  the  minor  premiss  in 
each  of  the  first  three  figures  of  syllogism  is  omitted. 

'Mr  X  is  a  profiteer,  and  therefore  he  ought  to  be 
super- taxed,' 

this  argument  is  acceptable  on  condition  that  the  re- 
quired major  premiss  'All  profiteers  ought  to  be  super- 
taxed'  is  admitted. 

'All  bodies  attract,  therefore  the  earth  attracts/ 

this  requires  the  minor  premiss  'The  earth  is  a  body.* 


102 


CHAPTER  IV 


*Mr  X  ought  not  to  be  super-taxed,  therefore  he  is 
not  a  profiteer,' 

this  requires  the  same  major  premiss  as  in  the  first 
example. 

'Everyone  present  voted  for  Home  Rule,  therefore 
Mr  Carson  was  not  present/ 

This  requires  as  minor  premiss  *Mr  Carson  would  have 
voted  against  Home  Rule.' 

'Mr  Carson  was  present,  therefore  someone  there 
must  have  voted  against  Home  Rule.' 

This  requires  the  major  premiss  'Mr  Carson  would  vote 
against  Home  Rule.' 

'The  earth  is  not  self-luminous,  therefore  not  all 
attracting  bodies  are  self-luminous.' 

This  requires  as  minor  premiss  'The  earth  is  an  attracting 
body.'  These  three  pairs  of  arguments  are  respectively 
in  the  first,  second  and  third  figure  of  syllogism. 

§  19.  Having  restricted  my  technical  treatment  of 
the  syllogism  to  a  single  chapter,  it  will  be  easily  in- 
ferred that  I  attach  considerable  importance  to  this  form 
of  inference,  while  at  the  same  time  I  hold  it  to  be 
only  one  among  many  other  equally  important  forms  of 
demonstrative  deduction.  Syllogism  is  practically  im- 
portant because  it  represents  the  form  in  which  persons 
unschooled  in  logical  technique  are  continually  arguing. 
It  is  theoretically  important  because  it  exhibits  in  their 
simplest  guise  the  fundamental  principles  which  underly 
all  demonstration  whether  deductive  or  inductive.  1 1  is 
educationally  important  because  the  establishment  of  its 
valid  moods  and  the  systematisation  and  co-ordination 
of  its  rules  afford  an  exercise  of  thought  not  inferior  and 
in  some  respects  superior  to  that  afforded  by  elementary 
mathematics. 


K. 


s 


I. 


CHAPTER  V 

FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM 

§  I.  The  categorical  syllogism  treated  in  the  last 
chapter  is  correctly  described  as  subsumptive.  This 
term  applies  strictly  to  the  first  figure  alone — which 
may  be  called  the  direct  subsumptive  figure,  and  since, 
either  by  antilogism  or  by  conversion,  the  other  iigures 
can  be  reduced  to  the  first,  these  may  be  called  indi- 
rectly subsumptive  figures.  As  explained  in  Chapter  I, 
this  form  of  inference  employs  in  the  most  biniple 
manner  the  Applicative  followed  by  the  Implicative 
principle.  The  ordinary  subsumptive  syllogism  has  a 
conclusion  applying  to  the  same  range  as  the  instantial 
minor,  and  its  typical  form  is : 

Everything  is/  if  w  ; 
This  is  m  \ 
.*.  This  is/. 

The  first  step  in  the  extension  of  the  ordinary  syllogism 
to  its  functional  form  is  to  take  a  conjunction  of  dis- 
connected syllogisms  of  the  type  : 

'  Everything  is/  \{  m-,  This  is  ;;^ ;  .  *.  This  is/.' 

*  Everything  is  /'  if  m* \  This  is  ;;^';  .  *.  This  is/'.' 

'  Everything  is/'' if  ;;^'';  This  is  ;;^";  .*.  This  is/",' etc. 

We  next  take  m,  m\  m",  etc.,  to  be  determinates 
under  the  determinable  M,  and/,  /',  /",  etc,  to  be 
determinates  under  the  determinable  P.  If  then  we 
can  collect  these  major  premisses  into  a  general  formula 
holding  for  every  value  o^  M  and  P  m 


accordance  with 


■'«■ 


104 


CHAPTER  V 


the  mathematical  equation  P=f{M)y  then  we  have  an 
example  of  what  may  be  called  the  functional  exten- 
sion of  the  syllogism,  or  (more  shortly)  of  the  functional 
syllogism,  where  the  major  or  supreme  premiss  may 
be  expressed  in  the  simple  form  P=^f{M),  Thus  in 
the  subsumptive  syllogism  the  terms  that  occur  in  the 
minor  and  major  premisses  are  merely  repeated  in  the 
conclusion ;  but,  in  the  functional  syllogism  which  yields 
an  indefinite  number  of  different  conclusions  for  the 
different  minors,  the  terms  which  occur  in  these  dif- 
ferent minors  and  conclusions  are  specific  values  of 
the  determinables  presented  in  the  supreme  premiss. 
Now  it  will  be  seen  that  no  other  principles  are  used 
in  the  functional  syllogism,  except  the  Applicative  and 
Implicative,  which  together  are  sufficient  to  extend 
deduction  beyond  the  scope  of  merely  subsumptive 
syllogism.  As  a  concrete  example,  let  us  take  the 
formula  of  gravitation,  which  may  be  elliptically  ex- 
pressed *  Acceleration  P  varies  inversely  as  the  square 

of  the  distance  J//  and  written  in  the  form  P=  -jrr^. 

Then,  by  the  Applicative  Principle  : 

'Ifi^=7,  t\i^nP  =  ^: 
and  adding,  as  Minor  Premiss  :  - — 

*  In  this  instance  M=^  7,' 
we  infer,  by  the  Implicative  Principle : 

*  In  this  instance  P  —  -^.' 

Similarly  when  the  value  oi  M  is  11,  the  value  of 
P  will  be  xir,  and  so  on.  The  same  form  of  inference 
holds  for  two  or  more  independent  variables :    thus 


s 


\t 


M 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    105 

Boyle's  Law  may  be  written  '  T=2^gPy";  then,  as 
before,  we  infer : 

When  the  value  of  P  is  5,  and  the  value  of  V  is  2, 
the  value  of  7"  will  be  2390. 

When  the  value  of  P  is  3,  and  the  value  of  V  is  7, 
the  value  of  7^  will  be  5019. 

§  2.  A  functional  expression  is  of  course  familiar 
to  the  mathematician,  but  it  will  be  important  to  ex- 
amine the  logical  principles  in  accordance  with  which  a 
universal  functional  formula  operates  in  mathematical 
demonstration.  In  the  first  place  we  may  observe  that, 
as  in  ordinary  syllogism,  the  supreme  or  major  tu no- 
tional universal  must  always  have  been  ultimately 
established  by  means  of  inductive  generalisation,  and 
in  the  last  resort  from  intuitive  or  experiential  data. 
Further,  the  functional  universal  may  be  said  to  be  a 
universal  of  the  second  order,  because  it  not  only 
universalises  over  every  instance  of  a  given  value  m^ 
but  applies  also  to  every  value  of  M,  In  deducing 
from  the  major  'P=/{My  conjoined  with  the  minor 
*A  certain  given  instance  is  m/we  reach  the  conclusion 
*This  given  instance  is  /,'  where  /  is  found  from  the 
equation  'p-f{ni)!  Here  it  is  to  be  observed  :  first, 
that  this  type  of  conclusion  can  be  drawn,  not  only  for 
the  minor  which  predicates  in,  but  also  lor  minors  which 
predicate  any  other  value  oi"  M ;  and  secondl}%  that  the 
character  predicated  in  each  conchision  is  not  merely 
what  is  predicated  in  the  functional  mcijor,  but  a  deter- 
minate  specification  of  this  predicate. 

In  the  functional  s\ih')ijisms  that  we  shall  consider 
in  this  chapter,  the  functional  nicuor  is  to  be  understood 
to  express  a  factual  rule,  or  more  particularly  a  Law  of 


io6 


CHAPTER  V 


Nature.    The  general  conception  of  a  Law  of  Nature 
has  been  discussed  (in  Chapter  XIV  of  Part  I)  under 
the  head  of  the  Principles  of  Connectional  Determina- 
tion.   There  it  is  shown  that  a  typical  uniformity  or 
Law  of  Nature  may  be  expressed  in  the  form  that  the 
variations  of  a  certain  phenomenal  character  depend 
upon  an  enumerable  set  of  other  phenomenal  characters  ; 
of  these  the  former  is  taken  to  be  connectionally  de- 
pendent upon   the   others,   which   are   connectionally 
independent  of  one  another.   A  specific  universal,  which 
expresses  such  a  relation  of  dependence  may  also  be 
called  a  Law  of  Covariation;    for  the  nature  of  the 
dependence  (say)  of  P  upon  ABC  is  such  that  all  the 
possible  variations  of  which  Pis  capable  are  determined 
by  the  joint  possible  variations  of  A,  B,  C,  which  are 
themselves  connectionally  independent  of  one  another. 
§  3.    We  have  a  special  case  of  this  relation  of  de- 
pendence or  covariation  when  the  determined  character 
can  be  represented  as  a  mathematical  function  of  the 
determining  characters  ;  and  it  is  this  special  case  which 
gives  rise  to  the  functional  syllogism.    Now,  in  a  func- 
tional major  expressed  (say)  in  the  form  P=f{A,  By  C), 
it  may  in  general  be  assumed  that  the  correlation  of 
these  variables  is  such  that,  not  only  can  the  value  of 
P  be  calculated  from  any  assigned  values  of  A,  By  C\ 
but  also,  conversely,  that  the  value  of  A  can  be  cal- 
culated from  the  values  of  PyByC\  and  that  of  B  from 
the  values  of  Py  Ay  C ;  and  that  of  C  from  the  values 
of  Py  A,  B;  and  similarly  for  a  larger  number  of  such 
connected  variables.  This  process  is  expressed  in  mathe- 
matical terms  as  solving  the  equation  P=^/{Ay  B,  C), 
to  find  the  value  of  Ay  which  is  thus  calculated  as  a 


11 


r 


n.^ 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    107 

certain  function  of  P,  By  C,  and  so  on.  A  convenient 
symbolisation  for  these  several  equations  will  be  as 
follows : 

from  which  we  calculate 

A  =MP,  B,  C) ;  B=/,iP,  A,  C) ;  C=/(/>,  A,  B). 

It  is  here  assumed  not  only  that  P  (in  mathematical 
phraseology)  is  a  single-valued  function  of  A,  B,  C, 
but  also  that  in  solving  this  equation  to  determine  A 
or  B  or  C  respectively,  these  also  are  single-valued 
functions  of  the  remaining  variables.  When  this  as- 
sumption holds,  we  may  speak  in  a  special  sense  of  the 
reversibility  of  cause  and  effect ;  i.e.  not  only  is  the 
effect  P  uniquely  determined  by  the  conjunction  of  the 
cause-factors  Ay  By  Cy  but  also  each  of  the  cause- 
factors  themselves,  such  as  Ay  is  uniquely  determined 
by  the  effect-factor  P  in  conjunction  with  the  remaining 
cause-factors  B  and  C  In  the  simplest  cases  reversi- 
bility follows  immediately  from  \k\^  form  of  the  function 
as  seen  in  the  example  given  of  Boyle's  Law.  Here 
we  have  a  correlation  between  temperature,  pressure, 
and  volume,  in  which  a  constant,  say  ky  is  involved, 
and  which  assumes  indifferently  the  form : 

^  pv  ke       ^    ke 

0=^-Y  y  or  v  =  —-y  or  p  =  — . 
k  p  ^       V 

In  this  simple  case,  the  multiplier  k  indicates  the 
special  form  of  the  function  which  in  the  general  case 
was  represented  by  the  unassigned  but  constant  symbol 
f.  An  equation  which,  in  this  way,  solves  uniquely  for 
all  the  variables  is  known  as  linear.  But  even  in  the 
case  of  non-linear  equations  we  must  be  able  to  deter- 


io8 


CHAPTER  V 


mine,  amongst  the  theoretically  possible  solutions  for 
any  one  of  the  variables,  that  which  is  the  sole  factual 
value.  In  other  words,  a  unique  determination  of  all 
the  variables,  in  terms  of  a  given  number  of  them,  may 
be  taken  as  expressing  the  actual  concrete  fact. 

§  4.  In  this  connection  it  is  important  to  note  the 
number  of  variables  entering  into  the  functional  formula. 
In  Boyle's  Law  this  number  is  three;  i.e.  there  are 
three  variables,  any  one  of  which  is  connectionally 
dependent  upon  the  two  remaining  variables,  so  that 
the  scope  of  dependence  may  be  measured  either  as 
two  or  as  three :  for  the  functional  formula  contains 
three  variables  which  are  notionally  independent  of  one 
another,  namely  p,  v,  6  \  but  of  these  two  only  are 
connectionally  independent  of  one  another.  These  two 
may  be  taken  indifferently  either  as  v  and  ^,  or  as  / 
and  Q,  or  as  /  and  v,  where,  according  to  the  alternative 
taken,  /  or  z;  or  ^  is  connectionally  dependent  upon 
the  two  others.  In  general,  when  there  are  afunctional 
relations,  connecting  n  notionally  independent  variables, 
then  any  n  —  r  of  these  can  be  taken  as  connectionally 
independent  of  one  another,  and  each  of  the  remain- 
ing r  as  connectionally  dependent  jointly  upon  the 
others. 

Thus  when  «  =  8,  and  ^  =  3,  the  three  functional 
relations  may  be  symbolised  : 

p=/^{aj,c,d,e);  g=/^{a,6,  c,  cl,  e);  r=/r{a,  d,  c,  d,e); 

or  adopting  a  shorter  notation  : 

p  =^fp .  abcde  ;  q  =y^ .  abcde ;  r  —f^ .  abcde. 

Such  a  trio  of  equations  are  taken  to  be  implicationally 
independent  of  one  another ;  i.e.  from  neither  one  or 


t 


1 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    109 

two  of  them  could  we  infer  the  third.  Otherwise  the 
number  three  would  reduce  to  two.  Now  the  number 
of  implicationally  independent  equations  is  necessarily 
the  same  as  the  number  of  connectionally  dependent 
variables.  Hence  for  the  case  under  consideration  we 
may  express  the  three  independent  functional  relations 
in  either  of  four  typical  forms  : 


1.  p  ^  fp  '  abcde  \ 

2.  a^fa-pbcde-, 

3-  P'^fp'  abcqr\ 

4.  a—  fa- pqrde\ 


q  "^^  fq'  abcde 'y 
q  =/q  .pbcde\ 
d=fd  •  abcqr\ 
b^fi,.pqrde\ 


r  ^fr .  abcde. 
r  =fr  ,  pbcde. 
e  —  fe .  abcqr. 
c  —  fc '  pqrde. 


The  first  trio  expresses  the  three  effect-factors  sepa- 
rately in  terms  of  the  five  cause-factors  jointly  ;  the 
second  expresses  one  cause-factor  and  two  effect- 
factors  separately  in  terms  of  one  effect-factor  and  four 
cause-factors  jointly ;  the  third  expresses  one  effect- 
factor  and  two  cause-factors  separately  in  terms  of 
three  cause-factors  and  two  effect-factors  jointly ;  and 
the  fourth  expresses  three  cause-factors  separately 
in  terms  of  three  effect-factors  and  two  cause-factors 
jointly. 

In  illustration  of  this  general  principle  we  will  con- 
sider the  Law  of  Gravitation,  which  may  be  formulated 


A^c 


m^m^ 


where  A  is  the  force  of  attraction  of  any  two  masses 
m^  and  m^  whose  distance  is  d^  c  being  constant  for  all 
variations  of  m^  and  m^y  as  well  as  of  d.  In  any  appli- 
cation of  the  above  formula  we  must  first  suppose  m^ 
and  m^  to  be  constant,  so  that  the  variation  of  A  de- 
pends solely  upon  that  of  d.  The  algebraical  equation 
here  is,   however,  logically  incomplete.     In   the    lirs^t 


no 


CHAPTER  V 


place,  as  regards  the  effect  A,  we  must  add  the  state- 
ment that  it  is  a  force  acting  in  the  direction  of  the 
line  joining  m^,  m^.  In  the  second  place,  as  regards 
the  cause  d,  not  only  must  the  distance  of  the  line 
joining  m^  and  m^  be  taken  as  a  cause-factor,  but  also 
its  direction. 

In  comparing  the  Law  of  Gravitation  with  Boyle's 
Law,  the  constants  k,  c,  m^,  m^,  represent  unchange- 
able properties  of  the  bodies  concerned,  while/,  dy  v,  d 
represent  their  changeable  states  or  relations.  It  is 
necessary  then  to  include  amongst  the  independent 
cause-factors  the  permanent  properties  of  bodies  as 
well  as  their  alterable  states  or  relations. 

§  5.  In  our  typical  expression  of  a  set  of  functional 
equations  the  number  of  variables  taken  to  be  con- 
nectionally  independent  was  the  same  in  all  the  several 
equations.  But  a  very  important  type  of  connectional 
formulae  is  that  in  which  equations  enter  involving 
different  numbers  of  independent  variables.  Consider 
the  following : 

Let  a  body  be  allowed  to  fall  in  vacuo.  Here  the 
two  independent  cause-factors  are  the  mass  (m)  of  the 
body  and  the  distance  (cl)  from  which  it  falls  to  the 
earth.  The  effect-factors  to  be  considered  are  the  time 
{t)  of  falling,  and  the  impulse  (/)  of  the  body  upon  the 
earth.  Since  out  of  the  four  variables  m,  d^p,  /,  two  of 
them,  namely  m  and  d,  are  (as  cause-factors)  connec- 
tionally  independent,  the  standard  form  in  which  both 
of  these  would  enter  into  the  function  is 

p  —fp .  md  and  /  =f^ .  md. 

But,  where  the  body  falls  in  vacuo,  the  time  (/)  is  in- 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    iii 

dependent  of  the  mass  {my.    Hence  in  this  case  the 
two  formulae  assume  the  form 

p^fp.  md ^nd  t=/^.d. 
In  this  case,  since  the  solution  of  the  equations  gives 
uniquely  determined  roots,  we  have  : 

(0/=//-^^»  (2)w=/«.M  {Z)d^f^,mp 
and  (4)  t^f,.dy  {s)d=^Mt)y 

and,  by  substitution  from  (5)  in  (i),  (2),  (3)  respectively, 

{^)P^fp^m  {7)m=f^,pt,  {%)t=/,.mp. 
Now,  since,  of  the  four  variables  m^  d,  /,  /,  any  two 
except  t  and  d  may  be  taken  as  connectionally  inde- 
pendent, either  one  of  the  following  pairs  of  connec- 
tional equations  may  be  used,  thus : 


Taking  m  and  ^as  independents  :  p  —f.  md 

m  and  /    „  „  :  p^f.mt 

p  and  ^  „  „  \  m  —f.pd 

p  and  /   „  „  \  m  ^  f.pt 

m3indp  „  „  :    d^f.mp 


>» 


»» 


» 


» 


with  t^f,d, 
d^f.t, 
t^f.d, 
d^f.t, 
t^f,  mp. 


Giving  to  the  unassigned  functions  their  actual  form, 
we  have  here 

{^)  p  =  mgt  ^nd  (5)^=i^/^, 
where  the  constant  g  stands  for  the  acceleration  32  ft. 
per  second.  Solving  these  equations  so  as  to  express  the 
effect-factors  p  and  /  in  terms  of  the  cause-factors  m  and 
dy  we  have 


V' 


§  6.    The  example  just  given  suggests  a  certain 
further   characteristic   of  the   connectional   equations 

^  This  illustrates  the  principle  underlying  the  inductive  method 
of  agreement ;  where  m  is  eliminated  as  a  cause-factor  relative  to  the 
effect  /,  since  a  variation  in  m  does  not  entail  a  variation  in  /. 


112 


CHAPTER  V 


of  applied  mathematics.  As  will  be  seen  in  the  above 
illustration,  the  connectional  equations  from  which  the 
deductive  process  derives  other  but  equivalent  equations 
are  of  a  mixed  nature  as  regards  the  variables  that 
are  taken  as  independents.  Of  the  two  equations 
p=imgly  and  d=^£-^\  from  which  the  other  equations 
are  derived,  the  former  expresses  the  effect-factor  /  in 
terms  of  the  cause-factor  m  and  the  other  effect-factor  / ; 
while  the  second  expresses  the  cause-factor  d  in  terms 
of  the  effect-factor  L  It  is  therefore  necessary  to  solve 
this  pair  of  equations  by  an  appropriate  process  in 
order  to  derive  the  pair  of  equations  which  express  the 
effect-factors  in  terms  of  the  cause-factors ;  namely  in 
the  form 

What  holds  in  this  particular  example  may  be  gene- 
ralised. Instead  of  separating  variables  that  are  given 
from  those  which  have  to  be  deduced,  we  have  a  set  of 
equations  (corresponding  in  number  to  the  dependent 
variables)  which  all  the  variables  taken  together  have 
to  satisfy.  Thus,  in  the  above  example,  the  equa- 
tions were  not  at  first  expressed  by  taking  a  pair  of 
cause-factors  as  independent  upon  which  the  pair  of 
effect-factors  depended,  but  the  first  of  the  two  equa- 
tions was  taken  from  the  pair  in  which  m  and  t  were 
supposed  to  be  independent,  and  the  other  from  the 
pair  in  which  m  and  /,  or  p  and  /  were  taken  as  indepen- 
dent. That  the  particular  example  of  the  falling  body, 
originally  taken  to  illustrate  a  different  principle,  should 
have  lent  itself  to  the  principle  now  under  considera- 
tion, is  more  or  less  accidental,  and  we  will  now  put 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    113 

forward  an  example  which  more  naturally  exhibits  this 
new  speciality  of  a  set  of  determining  equations. 

Thus:  consider  the  effect  of  mixing  two  substances 
at  different  temperatures  in  order  to  find  the  resultant 
temperature  which  will  be  reached  when  thermal  equi- 
librium has  been  established.     Here  we  must  take  as 
the  causally  determining  factors,  the  two  initial  tem- 
peratures ^„  ^2,  and  the  two  thermal  capacities  of  the 
substances  k^  and  k,^.    The  factors  to  be  determined  are 
the  heat  H^  entering  into  or  passing  from  the  one  sub- 
stance, and  the  heat  H^  passing  from  or  entering  into 
the  other,  together  with  the  final  temperature  0.    Now 
the  equations  that  must  be  here  used  express  the  con- 
ditions that  are  to  be  satisfied— the  effects  not  being, 
in  the  first   instance,  expressible  as  functions  of  the 
cause-factors.     These  equations  of  condition  are  the 
following  three : 

from  which  we  find 


0  = 


4  +  4 


'  ~"4+^r-'  ^^ — 4Tir~- 

The  solutions  of  these  equations  give  the  three  values 
Hy  H^y  and  0  (respectively)  that  were  to  be  deter- 
mined. Thus,  in  the  final  solution  we  have  succeeded 
in  expressing  the  factors  to  be  determined  in  terms  of 
the  determining  factors.  But,  in  the  equations  express- 
ing  the  conditions  to  be  satisfied,  the  first  two  express 
an  effect-factor  as  a  function  of  two  of  the  cause-factors 
and  one  of  the  effect-factors,  and  the  third  equation 


J.  L.n 


8 


^ 


N 


\ 


114 


CHAPTER  V 


expresses  one  effect-factor  as  a  function  of  another 
effect-factor.  The  use  of  equations  of  this  kind  is 
necessitated  by  the  inadequacy  of  our  knowledge  of 
the  precise  temporal  process  by  which  the  causal  con- 
ditions operate  until  the  final  issue  is  reached.  Thus,  in 
the  actual  process,  heat  will  be  passing  to  and  fro  from 
one  to  the  other  of  the  two  substances,  and  this  will 
entail  a  rise  or  fall  of  their  temperatures  in  an  in- 
calculable way,  which  may  be  roughly  expressed  by 
suggesting  that  the  quantum  of  heat  entering  the 
cooler  body  may  be  too  great,  so  that  the  flow  of  heat 
will  immediately  be  reversed;  and  this  process  might 
be  conceived  as  involving  even  an  infinite  number  of 
ingoings  and  outgoings  of  heat.  What  we  know,  how- 
ever, is  that  at  any  stage  of  the  process  the  heat  that 
leaves  one  body  must  be  equal  to  the  heat  that  enters 
the  other,  whether  this  quantum  is  to  be  reversed  in  the 
next  stage  or  not.  It  is  this  law  which  is  expressed  in 
our  third  equation,  while  the  other  two  equations 
express  a  law  or  property,  specific  to  the  two  substances, 
which  correlates  the  effect  upon  the  temperature  with 
the  quantum  of  heat  which  enters  or  leaves  the  body. 
What  then  we  know,  are  these  conditions  of  conserva- 
tion of  the  total  heat,  and  the  several  thermal  capacities 
of  the  bodies,  and  from  this  knowledge  the  final  effects 
can  be  calculated.  It  would  appear,  in  fact,  that  the 
cases  in  which  this  logical  principle  is  exhibited  are 
those  in  which  we  know  what  is  entailed  in  a  final 
state  of  equilibrium,  without  having  adequate  know- 
ledge for  tracing  in  detail  the  perhaps  oscillating  pro- 
cesses which  take  place  in  the  lapse  of  time  before  the 
final  state  of  equilibrium  is  reached. 


-^  f. 


s\ 


f 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    115 

We  have  illustrated  this  in  the  simplest  case,  where 
only  two  substances  are  mixed,  but  the  reader  will 
easily  be  able  to  construct  the  corresponding  equations 
for  any  given  number  of  different  substances.  In  all 
cases  the  final  or  resultant  temperature  is  equivalent  to 
the  arithmetic  mean  of  the  initial  temperatures,  each 
*  weighted'  by  the  corresponding  thermal  capacity. 
Thus 

A +  ^3  +  ...+^,. 
Now  having  given  an  illustration  from  physics,  we 
will  give  a  closely  analogous  illustration  from  economics. 
The  formula  of  covariation  which  connects  the  quantity 
of  a  commodity  that  is  exchanged  with  \X.s^  price  \^  such 
that  the  two  opposed  parties  shall  be  satisfied  at  the 
rate  of  exchange  finally  agreed  upon.  Now  the  formula 
of  covariation  on  the  side  of  demand  is  assumed  to  be 
connectionally  independent  of  that  on  the  side  of  supply. 
That  which  represents  the  economic  attitude  of  the 
consumers  depends  solely  upon  their  relative  desires 
for  different  commodities,  their  monetary  resources, 
and — we  may  add — the  prices  at  which  they  are  able 
to  buy  commodities  other  than  that  under  consideration. 
In  the  same  way,  the  attitude  of  the  producers  is  wholly 
independent  of  that  of  the  consumers;  and  depends 
upon  the  contract-prices  current  for  the  employment  of 
the  several  agents  of  production,  and  upon  the  efficiency 
of  these  agents  when  co-operating  in  producing  the 
commodity.  It  will  thus  be  seen  that  the  several 
factors  that  determine  the  conditions  of  supply  are 
independent  of  those  that  determine  the  conditions  of 
demand.    Here,  as  in  the  case  of  thermal  equilibrium, 

8—2 


ii6 


CHAPTER  V 


the  equations  of  condition  express,  not  the  effect-factors 
as  functions  of  the  cause- factors,  but  the  conditions 
taken  together  which  satisfy  the  consumers  and  the 
producers  regarded  each  as  economically  independent 

of  the  other. 

This  economic  illustration  differs  from  the  case  of 
thermal  equilibrium  in  the  important  respect  that  the 
two  functions  of  demand  and  supply  respectively  replace 
the  actually  operating  cause-factors,  which  are  highly 
complex  and  do  not  explicitly  enter  into  the  equations 
to  be  solved. 

§  7.  The  above  illustrations  of  the  functional  ex- 
tension of  the  syllogism  have  shown  how,  by  the  use 
of  a  set  of  functional  premisses  standing  as  majors,  we 
may  take  not  only  minors  which  enable  us  to  infer 
an  effect-factor  from  the  knowledge  of  a  given  cause- 
factor,  but  also  minors  which  enable  us  to  infer  a  cause- 
factor  from  the  knowledge  of  a  given  effect-factor.  The 
supposition  upon  which  this  is  based  has  been  called 
the  Principle  of  Reversibility.  We  shall  now  show  that 
it  is  this  principle  which  underlies  the  so-called  method 
of  Residues,  and  other  similar  deductive  processes.  The 
canon  of  this  method  is  stated  by  Mill  as  follows: 

'Subduct  from  any  phenomenon  such  part  as  is 
known  by  previous  inductions  to  be  the  effect  of  cer- 
tain antecedents,  and  the  residue  of  the  phenomenon  is 
the  effect  of  the  remaining  antecedents.' 

In  using  the  term  *  subduct'  Mill  intends  no  doubt  to 
hint  that,  in  the  simplest  cases,  for  'subduct'  we  may 
substitute  *  subtract.'  Thus  Jevons,  in  his  Elementary 
Lessons,  takes  the  case  of '  ascertaining  the  exact  weight 
of  any  commodity  in  a  cart  by  weighing  the  cart  and 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    117 

load,  and  then  subtracting  the  weight  of  the  cart  alone, 
which  has  been  previously  ascertained.'  Here  what 
corresponds  to  the  effect  (/)  is  the  weight  of  the  cart 
and  load  together,  and  its  causes  are  {a)  the  weight  of 
the  commodity,  and  {b)  the  weight  of  the  cart :  so  that 
the  functional  datum  assumes  its  simplest  form,  viz. 
p  =  a  +  b,  which  by  reversibility  gives  a^^p  —  b.  This  is 
a  case  of  solving  an  equation  /  =/(^,  b)  to  find  a,  and 
deducing  ^=y"(^,/),  the  equations  being  linear.  The 
next  simplest  example  of  such  reversibility  is  that  of  the 
composition  of  forces.  Here  the 
diagonal  OP  represents  the  effect, 
and  the  sides  OA,  OB,  the  cause- 
factors.  Just  as,  given  the  two 
cause-factors  OA  and  OB,  we 
drawy^P  parallel  and  equal  to  OB 
to  find  the  effect  0P\  so,  given  OA  as  one  cause-factor 
and  OP  as  effect,  we  may  draw  OB  parallel  and  equal 
to  AP  to  find  the  other  cause-factor  OB.  Innumerable 
other  examples  may  be  given  of  reversibility  for  more 
or  less  complicated  cases.  But  the  classical  example 
most  frequently  cited  is  the  Adams- Le verier  discovery 
of  the  planet  Neptune  from  the  observed  movements 
of  Uranus.  Here  we  may  represent  the  positions  and 
masses  of  the  Sun,  of  the  Moon  and  of  the  unknown 
Neptune  by  the  symbols  a,  b,  c  respectively;  and  the 
movement  of  Uranus  by  the  symbol  /.  Thus  p  was 
theoretically  known  as  a  given  function  of  a,  b,  c,  say 
/  =j//  (^»  b,  c),  where  /  stands  elliptically  for  the  effect, 
and  a,  b,  c  for  the  several  cause-factors.  The  solution 
for  c  was  then  uniquely  calculated  in  the  form 


\ 


\ 


ii8 


CHAPTER  V 


Now  it  will  be  observed  that  the  so-called  Method 
of  Residues,  which  is  based  upon  the  assumption  of 
reversibility  is  purely  deductive,  in  that  (i)  it  employs 
only  the  Applicative  and  Implicative  principles  of  in- 
ference, and  (2)  the  conclusion  obtained  applies  solely 
to  the  specific  instances  for  which  the  calculation  is 
made.  This  consideration  shows  that  there  is  no  justifi- 
cation for  putting  Herschel's  method  of  Residues  under 
the  head  of  methods  of  induction,  along  with  such 
methods  as  those  of  Agreement  and  Difference;   for, 
on  the  grounds  above  alleged,  it  is  purely  deductive. 
On  this  matter  Mill  sees  half  the  truth  ;  for,  in  com- 
paring the  Method  of  Residues  with  that  of  Difference, 
he  remarks  that  the  negative  instance  in  the  former  is 
not  the  direct  result  of  observation,  but  has  been  arrived 
at  by  deduction.    And  again,  in  his  formulation  of  the 
Canon  of  Residues,  he  speaks  of  'such  part  of  the 
phenomenon  as  is  known  by  previous  induction,*  where 
he  fails  to  note  that  what  is  known  by  previous  induction 
functions  merely  like  the  major  premiss  of  a  syllogism, 
and  therefore  does  not  in  any  way  render  the  inference 
inductive.  What  holds  for  the  method  of  Residues  holds 
also   of   many  less   technical   processes  which,  while 
purely  deductive,   have  been  obscurely  conceived  as 
inductive.    For  instance,   the  procedure  in  a  judicial 
enquiry  or  by  a  police  detective  or  of  historical  research 
in  discovering  the  specific  cause  of  a  complicated  set 
of   circumstances   constituting  an   observed  effect,   is 
purely  deductive ;    for  it  employs  as   major  premiss 
known  laws  of  human  or  physical  nature  under  which  the 
known  circumstances  are  to  be  subsumed  in  the  minor ; 
while  the  conclusion  refers  solely  to  the  case  sub  judice. 


»■ 


t 


I 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    119 

§  8.    Something  should  be  said  in  explanation  of  the 
fact  that  inferences  of  this  kind  are  so  frequently  spoken 
of  as  inductive.    It  is  not  only  because  the  major  premiss 
must  itself  have  been  obtained  by  induction,  but  further 
because  the  minor  premiss  represents  a  fact  obtained 
by  observation,  that  logicians  have  made  this  mistake ; 
for  the  notion  of  observation  or  experimentation  as  the 
method  by  which  new  knowledge  is  acquired  is   in- 
variably associated  with  induction.    But  it  should  be 
pointed  out  that  there  is  here  a  confusion  between  the 
matter  and  the  form  of  an  inference.    Mere  syllogism 
will  obviously  yield  new  material  knowledge,  provided 
that  the  minor  premiss  represents  new  material  know- 
ledge such  as  can  only  be  obtained  by  observation. 
For  example,  from  the  observation  that  the  importation 
of  food  has  been  taxed,  we  may  infer  the  new  material 
knowledge  that  the  price  of  food  will  rise  at  a  certun 
time  in  a  certain  economic  market,  if  we  have  been 
otherwise  assured  of  the  major  premiss  appropriate  to 
the  circumstance.    The  form  of  such  an  inference  is 
purely  deductive,  and  the  fact  that  historical  research— 
and   not  a  merely  foreknown  universal  formula     has 
been  required  to  establish  the  minor  does  not  rtnder 
the  argument  in  any  sense  inductive;  for  the  conclusion 
holds  only  of  the  period  and  region  to  which  the   causal 
occurrence  which  has  been  discovered  applies,  and  does 
not  involve  any  inductive  generalisation  from  one  period 
or  region  to  others.   A  further  explanation  of  this  com- 
mon error  is  to  be  found  in  the  fact  that  the  conclusion 
reached  deductively   for   a  given   instance   may  often 
be  verified  b>  awaiting  the  occasion  for  observing  the 
effect  in  that  instance.    Now  this  process  of  verilication 


120 


CHAPTER  V 


FUNCTIONAL  EXTENSION  OF  THE  SYLLOGISM    121 


merely  assures  us  that  we  have  adequately  estimated 
the  causes  operating  in  the  given  instance ;  but  it  has 
been  almost  invariably  confused  with  the  process  of 
verifying,  or  rather  confirming,  the  major  premiss  itself 
regarded  as  a  problematic  hypothesis  as  yet  unproven. 
§  9.  We  ought  now  to  distinguish,  in  these  functional 
extensions  of  the  syllogism,  the  element  which  is  purely 
subsumptive  from  that  which  is  functional ;  for  the  two 
elements  are  practically  always  united  in  any  concrete 
inference  of  the  functional  kind.  It  will  be  found  that 
the  factual  formulae  used  in  applied  mathematics  as 
major  premisses  for  deduction  necessarily  involve  two 
kinds  of  constituent,  one  of  which  is  known  as  variable 
and  the  other  as  constant.  The  mathematical  use  of  the 
Itrmconstant  presents  certain  difficulties  from  the  logical 
point  of  view.  There  are  certain  constants — e.g.  the 
specific  integers  and  the  algebraical  operators — which 
are  absolutely  constant  in  the  sense  that  in  all  their 
occurrences  they  stand  for  the  same  thing  and  are 
entirely  independent  of  context.  But  those  so-called 
constants  which  are  dependent  upon  context  are  only 
referentially  constant,  being  actually  variable  in  precisely 
the  same  sense  as  the  symbols  that  mathematicians 
recognise  as  variable.  To  explain  this  we  may  select 
illustrations  from  an  innumerable  variety  of  formulae 
used  in  applied  mathematics.  Consider,  for  instance,  the 
formula  which  expresses  the  elasticity  of  a  solid  body 
which  can  support  tension.  The  rule  upon  which  the 
extension  of  such  a  body  depends  is  shortly  expressed 
in  the  formula  T=kE,  where  7"  stands  for  the  variable 
tension  and  E  for  the  variable  extension;  while  k, 
which  is  said  to  be  constant,  stands  for  the  elasticity  of 


1 


the  particular  kind  of  solid  for  which  the  rule  holds. 
Now  this  coefficient  of  elasticity,  though  constant  for 
all  possible  variations  of  extension  and  tension  of  the 
body,  yet  varies  from  one  kind  of  solid  substance  to 
another.  We  shall  show,  then,  that  such  a  coefficient, 
which  mathematicians  call  constant,  is  used  in  the 
deductive  process  subsumptivelyy  while  that  which  is 
explicitly  regarded  as  variable  is  M^^di  functionally .  We 
may  mark  the  real  variability  of  a  so-called  constant  by 
a  subscript  indicating  the  specific  kind  of  substance  of 
which  the  coefficient  can  be  predicated.  Thus  k^  will 
stand  for  the  coefficient  of  elasticity  of  the  kind  of  sub- 
stance named  s ;  while  k^  (say)  will  stand  for  that  holding 
for  the  kind  of  substance  called  /.  To  express  the 
mathematical  procedure  in  strictly  explicit  logical  form : 

Major  Premiss,    Every  body,  say  b,  which  is  k,  has 
the  property  expressed  by  the  algebraical  equation 

Minor  Premiss,    A  certain  body  b  is  k,. 

Conclusion.    The  body  b  has  the  property  expressed 
by  the  equation  T—k^E, 

Now  this  is  a  merely  subsumptive  syllogism,  in  which 
the  coefficient  k^  and  the  body  b  recur  unmodified  in 
the  conclusion  as  in  the  premisses.  Thus,  the  coefficient 
which  is  called  constant  is  used  solely  in  a  subsumptive 
form  of  syllogism ;  but,  inasmuch  as  a  similar  formula 
applies  to  bodies  of  a  different  nature  (such  as  /),  the 
coefficient  k  is  not  absolutely  constant  but  varies 
according  to  the  substance  of  the  solid.  In  logical 
analysis,  we  must  recognise  the  distinct  ways  in  which 
the  so-called  constants  and  the  so-called  variables  enter 
into  the  deductive  process.    This  may  be  expressed 


122 


CHAPTER  V 


logically  by  defining  the  order  in  which  the  variations 
have  to  be  made.  For  we  \i^,v^  first  to  consider  varia- 
tions of  the  so-called  variables,  which  determine  the 
range  of  the  conclusion  as  holding  for  every  case  over 
which  the  constant  applies.  Only  after  this  range  of 
variation  has  been  taken  into  consideration  may  we 
proceed  to  vary  the  so-called  constants,  and  for  any 
new  value  carry  out  the  same  range  of  variations  of  the 
variables.  In  language  borrowed  from  mathematical 
terminology,  we  may  say  that  the  variations  of  the 
explicit  variables  are  to  be  made  within  the  bracket, 
while  the  variations  to  be  made  of  the  so-called  constants 
are  to  be  made  outside  the  bracket. 


/ 


K 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


§  I.  Under  this  heading  we  shall  discuss  the  prin- 
ciples underlying  the  deduction  of  formulae  in  the 
sciences  of  mathematics  and  logic.  Although  properlv 
speaking  pure  mathematics  is  a  development  of  logic, 
yet  certain  important  points  of  distinction  between  the 
two  sciences  must  be  brought  out.  It  has  been  vt  ry 
commonly  assumed  that  the  sole  method  of  deductive 
procedure  in  pure  mathematics,  including  Geometry, 
is  syllogistic.  Now  although  it  will  be  found  that  no 
fundamental  principle  is  employed  in  mathematical  cle- 
duction  other  than  the  Applicative — which  is  essential 
for  syllogism — yet  the  conclusions  successively  derived 
from  previously  established  formulae  are  not  such  as 
could  be  inferred  by  means  of  any  mere  chain  of  syl 
logisms.  To  explain  this,  it  is  necessary  to  point  out  the 
peculiar  nature  of  the  relation  between  conclusion  and 
premisses  in  mathematical  processes.  Ordinary  syl 
logism,  as  has  been  ex|:)lained,  is  of  the  compiiratively 
simple  type  denominated  subsumptive.  I  f  siibsumpti ve 
inferences  only  were  used  in  algebra  or  geometry,  it 
would  be  impossible  to  demonstrate  conclusions  except 
for  special  cases  subsumabie  under  the  primary  intuited 
axioms  or  under  some  previously  established  formulae. 
Thus  from  such  premisses  as:  'Everything  thcit  is  ;;/  is 
p'  and  'Everything  that  is  n  is  q'  we  could  infer  sub- 


'^i 


124 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


125 


sumptively  only  that  *  Everything  that  is  m  and  n\sp 
and  q'  In  other  words,  by  means  of  subsumptive  de- 
duction, we  can  infer  only  that  what  holds  universally 
of  the  members  of  a  genus,  77z  or  n,  holds  universally  of 
the  members  of  their  common  species,  viz.  of  the  things 
that  are  characterised  as  being  both  m  and  n.  For 
example:  in  geometry,  having  established  a  formula  for 
all  triangles  and  a  formula  for  all  right-angled  figures, 
we  could  by  merely  subsumptive  inference  predicate  of 
any  species  of  triangles — say  right-angled — only  what 
could  be  predicated  of  all  triangles ;  and  similarly  we 
could  predicate  of  any  species  of  right-angled  figures — 
say  three-sided — only  what  could  be  predicated  of  all 
right-angled  figures.  But  actually  in  geometry  we  prove 
a  property  (viz.  the  Pythagorean)  of  all  right-angled 
triangles  which  is  not  the  same  as  any  universal  pro- 
perty either  of  three-sided  or  of  right-angled  figures. 
Similarly  in  algebra,  we  can  deduce  properties  of  all 
integers  divisible  by  2  and  divisible  by  3,  which  hold 
neither  of  all  integers  divisible  by  2  nor  of  all  integers 
divisible  by  3.  A  predicate  which  holds  for  all  members 
of  a  species,  but  not  for  all  members  of  any  genus  to 
which  by  definition  the  species  belongs,  is  technically 
known  as  a  proprium  or  tStoi/,  either  of  which  term  may 
be  translated  property.  It  is  one  of  the  special  objects 
of  this  chapter  to  analyse  the  process  by  which  proper- 
ties, in  this  technical  sense,  are  deduced.  It  will  be 
shown  that,  in  the  deductions  peculiar  to  pure  mathe- 
matics, the  premisses  and  conclusions  assume  the  form 
of  functional  equations;  and  that  it  is  owing  to  this 
characteristic  that  properties  in  the  technical  sense  can 
be  deductively  demonstrated.    We  therefore  give  the 


W^^ 


mm^  functional  dedzution,  in  antithesis  to  subsumptive 
or  syllogistic  deduction,  to  the  specifically  mathematical 
form  of  inference. 

§  2.  Before  entering  upon  the  main  discussion  it 
will  be  well  further  to  consider  the  nature  of  the  Aris- 
totelean  Ihiov.  Many  modern  logicians  have  filled  to 
grasp  the  important  significance  to  be  attached  t  >  this 
notion.  Elementary  textbooks,  such  as  that  ot  j evens, 
define  a  property  of  a  class  as  any  character  noi  in 
eluded  in  the  connotation,  which  can  be  predicated  (  f 
all,  as  distinct  from  an  accident  which  can  be  predicated 
only  of  some,  members  of  the  class.  On  the  other  hand. 
Mill  attempts  to  define  a  proprium  in  closer  connection 
with  the  scholastic  development  of  Aristotle's  doctrine, 
and  distinguishes  not  merely  between  an  invariable  and 
a  variable  predicate  of  a  class — which  satisfies  Jevons — 
but  defines  a  proprium  as  a  predicate  not  mcluded  in 
the  connotation  of  the  class  (and  therefore  assertible  in 
a  proposition  not  merely  verbal)  but  following  neces- 
sarily from  the  connotation  alone.  But  since  a  pro- 
position  which  merely  asserts  connotation  is  verbal, 
this  account  of  the  proprium  is  incompatible  with  the 
theory — so  clearly  expounded  in  his  chapters  on  Deh- 
nition  and  on  Verbal  Propositions — that  no  conclusion 
can  be  drawn  from  merely  verbal  propositions  that  is 
not  itself  merely  verbal.  From  this  it  follows  that  in 
order  demonstratively  to  establish  any  invariable  charac- 
ter that  can  be  regarded  as  necessary,  we  require  as 
premisses  not  only  definitions  but  also  real  or  genunie 
propositions,  and,  in  mathematics,  ultimatelv  axioms. 
It  is  true  that  Mill  distinguishes  two  ways  in  which  the 
proprium  may  follow  necessarily  from  the  connotation: 


/ 


126 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


X27 


*it  may  follow  as  a  conclusion  follows  premisses,  or  it 
may  follow  as  an  effect  follows  a  cause/  But  this  dis- 
tinction is  purely  illusory  and  wholly  irrelevant  to  the 
notion  of  necessity  of  demonstration ;  for,  in  both  cases, 
the  ground  for  Mill's  account  of  a  proprium  as  neces- 
sarily following  from  the  connotation  is  that  appropriate 
knowledge  will  enable  us  to  infer  demonstratively  the 
proprium  from  the  connotation.  A  legitimate  distinc- 
tion may  be  drawn  according  as  the  major  premiss  from 
which  a  proprium  is  inferred  is  of  the  nature  of  an  axiom 
or  of  a  causal  law.  Indeed  Mill  himself  goes  on  to  say 
that  the  necessity  attributed  to  the  proprium  means  that 
*its  not  following  would  be  inconsistent  with' — i.e.  its 
following  could  be  inferred  from — either  an  Axiom  or 
a  Law  of  Nature.  Thus  in  both  cases  the  notion  of 
following  is  the  same,  and  simply  means  inferrible  from. 
The  proprium,  therefore,  never  follows  from  the  conno- 
tation alone,  but  requires  in  addition  one  or  other  of 
the  two  species  of  real  propositions,  axiomatic  or  ex- 
periential, to  serve  as  major  premiss. 

§  3.  The  functional  equations  used  in  the  deductions 
of  pure  mathematics  in  some  respects  differ  from  and 
in  others  agree  with  those  used  as  major  premisses  in 
the  process  discussed  under  the  head  of  the  functional 
extension  of  the  syllogism.  The  equation  used  in  this 
latter  process  serves  as  a  single  major  premiss  for  a 
number  of  specific  conclusions  found  by  replacing  the 
variables  by  their  specific  values.  Here  the  functional 
equation  assumes  the  form  P=f  (Ay  B,  C)  for  all  values 
ofAyB,  C,  But  the  equations  used  in  the  process  of  func- 
tional deduction  are  of  the  form/(y4,  B,C)  =  (f>{Ay  B,  C) 
for  all  values  of  A,  B,  C,  where  all  the  variables  are 


i.p.- 


I 


independendy  variable,  and  the  equation  therefore 
contains  no  such  symbol  as  P  that  can  be  exhibited  as 
dependent  upon  the  others.  The  distinction  between 
these  two  types  of  equation  is  familiar  to  mathematicians ; 
the  former  may  be  called  a  limiting,  the  latter  a  non- 
limiting  equation.  The  limiting  equation  is  generally 
used  to  determine  one  or  other  of  the  quantities  P.,  A, 
By  or  C,  in  terms  of  the  remainder,  ^o  that  liere  we 
associate  the  antithesis  between  dependent  and  inde- 
pendent with  the  antithesis  between  unknown  and 
known;  whereas,  in  the  non-limiting  equation,  no  one 
of  the  variables  can  be  regarded  as  unknown  and  as 
such  expressible  in  terms  of  the  others  regarded  as 
known.  The  distinctions  that  have  been  put  forward 
between  these  two  types  of  functional  process  are  tanta- 
mount to  defining  the  functional  syllogism  as  that  which 
proves  factual  conclusions  from  tactual  premisses,  and 
functional  deduction  as  that  which  proves  formal  conclu- 
sions or  formulae  from  formal  premisses,  i.e.  from 
formulae  previously  established.  Tt  wnll  further  be  ob- 
served, from  the  simple  illustrations  which  follow,  that 
whereas  the  functional  syllogism  requires  only  the  one 
functional  equation  that  serves  as  major  premiss,  the 
process  of  functional  deduction  will  necessarily  ifivolve 
a  conjunction  of  two  or  more  functional  equations,  all 
of  which    are,   as    above   explained,    formal    and    not 

factual. 

To  illustrate  the  general  formula  used  in  functional 

deduction,  viz. : 

/(^,      by       Cy        .,.)    =    (f>{ay       b,      Cy        ...) 

which  is  understood  to  hold  for  every  value  of  the 


I 


\ 


/ 


/ 


128 


CHAPTER  VI 


variables  A^  By  C,  ...,  we  may  instance  the  following 
elementary  examples : 


and 


axb  =  bxay 


both  of  which  involve  two  variables;  and  again 

(a'^b)-\-c  =a-\-{b-\-c) 

and  {a-{-b)xc  ={axc)-\-{bx  c), 

both  of  which  involve  three  variables.  The  last  three 
formulae  are  known  respectively  as  the  Commutative, 
the  Associative  and  the  Distributive  Law. 

§  4.  In  the  functional  equations  of  mathematics  it  is 
important  to  realise  the  range  of  universality  covered  by 
any  functional  formula.  This  range  depends  upon  the 
numberof  independent  variables  involved  in  the  formula, 
the  range  being  wider  or  narrower  according  as  the 
number  of  independent  variables  is  larger  or  smaller. 
For  example,  supposing  that  jr,  y,  2  have  respectively 
7,  5,  10  possible  values;  then  the  numberof  applications 
of  the  formula  involving  x  alone  is  7,  that  of  a  formula 
involving  x  and  y  alone  is  35,  and  that  of  a  formula 
involving  x  and  y  and  2  is  350.  And  in  general,  the 
number  of  applications  of  a  formula  is  equal  to  the 
arithmetical  product  of  the  numbers  of  possible  values 
for  the  variables  involved.  Now  the  number  of  possible 
values  of  any  variable  occurring  in  logical  or  mathe- 
matical formulae  is  infinite;  hence,  for  the  cases  re- 
spectively of  I,  2,  3...  variables,  the  corresponding 
ranges  of  application  would  be  00,  00^,  00  \..,  consti- 
tuting a  series  of  continually  higher  orders  of  infinity ; 
or  rather,  in  accordance  with  Cantor's  arithmetic,  each 
of  the  ranges  of  application  for  i,  2,  3  ...  variables  is  a 


I 


L    f 


•#  » 


^** 


FUNCTIONAL  DEDUCTION  129 

proper  part  of  that  for  its  successor,   although  their 
cardinal  numbers  are  the  same. 

Now  it  will  be  found  that,  in  inferences  of  the  nature 

of  functional  deduction,  the  derived  formula  may  have 

a  range  of  application — not  narrower  than  but — equal 

to  or  even  wider  than  that  from  which  it  is  derived. 

Thus  the  word  deduction  as  here  applied  does  not 

answer  to  the  usual  definition  of  deduction  (illustrated 

especially  in  the  syllogism)  as  inference  from  the  generic 

to  the  specific ;  although  the  only  fundamental  principle 

employed  in  the  process  is  the  Applicative,  according 

to  which  we  replace  either  a  variable  symbol  by  one  of 

its  determinates  or  one  determinate  variant  by  another. 

But  here  a  distinction  must  be  made  according  as  the 

substituted  symbol  is  simple  or  compound.    I  f  we  merely 

replace  any  one  of  the  simple  symbols  a,  by  c  by  some 

other  simple  symbol  we  shall  not  obtain  a  really  new 

formula,  since  the  formula  is  to  be  interpreted  as  holding 

for  all  substitutable  values,  and  hence  it  is  a  matter  of 

indifference  whether  we  express  the  formula  in  terms  of 

the  symbols  a,  by  Cy  (say)  or  of/,  q,r.   In  order  to  deduce 

new  formulae,  it  is  necessary  to  replace  two  or  more  simple 

symbols  by  connected  compounds. 

For  those  unfamiliar  with  mathematical  methods,  it 
should  be  pointed  out  that,  when  any  compound  symbol 
is  substituted  for  a  simple,  the  compound  must  be  en- 
closed in  a  bracket  or  be  shown  by  some  device  to 
constitute  a  single  symbolic  unit.  Though  we  may 
always  replace  in  a  general  formula  a  simple  by  a  com- 
pound symbol,  the  reverse  does  not  by  any  means  hold 
without  exception.  The  cases  in  which  such  substitu- 
tion is  permissible  have  been  partially  explained  in  the 
J.  L.  n  Q 


\ 


130 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


131 


chapter  on  Symbolism  and  Functions.  There  it  was 
shown  that,  if  a  formula  involves  such  compound 
symbols  or  sub-constructs  as  f{a,  b)y  f{c,  d)  etc.,  and 
only  such,  where  none  of  the  simple  symbols  used  in 
the  one  bracketed  sub-construct  occur  in  any  of  the 
others,  then  these  bracketed  functions  are  called  dis- 
connected. It  is  in  the  case  of  disconnected  functions 
that  free  substitutions  of  simple  symbols  for  the  com- 
pound are  permissible.  The  reason  for  this  is  that,  for 
the  notion  of  a  function  of  any  given  variants,  it  is 
essential  that  these  shall  be  variable  independently  of 
one  another.  Now,  when  the  different  sub-constructs 
or  bracketed  functions  are  connected  with  one  another 
through  identity  of  some  simple  symbol,  say  a,  it  is 
clear  that  we  cannot  contemplate  a  variation  of  one  of 
these  compounds  without  its  involving  a  variation  of  the 
other  connected  compounds.  Hence  we  should  be  vio- 
lating the  fundamental  principle  of  independent  varia- 
bility of  the  variants,  if  we  freely  substituted  for  such 
connected  compounds  simple  symbols  which  would  have 
to  be  understood  as  capable  of  independent  variation. 
Hence,  it  is  only  when  the  various  compounds  involved 
in  a  function  are  unconnected,  that  for  each  of  such 
compounds  a  simple  symbol  may  be  substituted. 

§  5.  Returning  to  the  problem  under  immediate  con- 
sideration, a  simple  illustration  from  algebra  will  show 
how,  by  making  appropriate  substitutions  in  a  given 
functional  formula,  we  may  demonstrate  a  new  formula. 
Thus,  having  established  the  formula  that  for  all  values 

of  X  and  y 

(i)   {x^ry)^{x--y)^^^f 

we  may  substitute  for  ;irand  jv,  respectively,  the  connected 


1 "' 

4 


*  h 


1 


III 


compounds  a-Vb  and a-b\  and  so  deduce  (by  means  of 
the  distributive  law  for  multiplication  etc.)  that  for  all 
values  of  a  and  b, 

(ii)    4^^  =  (^  +  ^)2-(a-.^)^ 

This  is  a  new  formula,  different  from  the  previous  one, 
because  the  relation  between  a  and  b  predicated  in 
(ii)  is  different  from  the  relation  between  x  and  y  pre- 
dicated in  (i).  Moreover  the  range  of  application  for 
(ii)  is  no  narrower  than  that  for  (i);  for  (i)  applies  for 
every  diad  or  couple  'x  toy,'  and  (ii)  for  every  diad  or 
couple  'a  to  b' ;  and  therefore  the  ranges  for  (i)  and  (ii) 
are  the  same.  Again,  if  we  have  established  the  Com- 
mutative, Associative,  and  Distributive  formulae  given 
above,  the  reader  will  see  that,  by  means  only  of  the 
Applicative  principle,  we  can  deduce  from  these  three 
formulae  what  is  in  fact  a  new  formula: 

{a-}'b){c'\-d)  =  ac  +  bc  +  ad+bd. 

In  this  case,  the  formula  deduced  has  a  wider  range  of 
application  than  any  of  the  formulae  from  which  it  is 
deduced.  For  the  premisses  for  this  deduction  involve 
respectively  2,  3  and  3,  independent  variables,  while 
the  conclusion  involves  4;  showing,  as  explained  in  the 
previous  paragraph,  that  the  range  of  application  oi  the 
conclusion  is  wider  than  that  of  even  the  widest  premiss. 
To  reach  a  conclusion  inclusive  of  and  wider  than  the 
premisses  is  in  general  considered  the  mark  of  an  in- 
ductive inference;  but  we  have  shown  by  the  above 
example  that,  wherethe  premisses  are  functional  foi  niulae 
involving  more  than  one  independent  variable,  the  mere 
employment  of  the  Applicative  principle  enables  us  to 
reach  a  formula  wider  than  any  of  the  premisses.    Now 

9—2 


132 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


133 


it  is  in  accordance  with  general  usage  to  define  deductive 
inference  as  that  which  employs  no  principles  but  the 
Applicative  and  the  Implicative.    In  the  purely  deduc- 
tive process  of  mathematics,  in  fact,  it  is  only  the  Appli- 
cative principle  that  is  required ;  and  pure  mathematics 
is  regarded  as  specially  typifying  the  power  of  mere 
deduction.    It  is  true,  however,  that  mathematicians 
have  employed  a  method  which  involves  also  the  Impli- 
cative principle,  viz.  what  has  always  been  known  under 
the  name  of  *  mathematical  induction.'    In  these  later 
days,  this  method  has  been  regarded  as  more  specifically 
characteristic  of  mathematics  than  any  other.    But  the 
line  of  distinction  between  induction  and  deduction,  in 
their  extended  potentialities  fordemonstrative  inference, 
cannot  be  drawn  on  any  logical  principle  that  would  be 
universally  accepted.    It  is  for  this  reason  that  I  have 
attempted  to  treat  in  one  large  division  of  my  Logic  all 
varieties  of  demonstrative  inference,  on  the  ground  that 
it  is  the  demonstrative  character  of  these  inferences  that 
brings  them  within  one  sphere,  and  that  the  distinction 
that  might  be  drawn  between  deductive  and  inductive 
demonstration  has  no  important  logical  significance  com- 
parable with  that  between  demonstrative  and  proble- 
matic inference.    Mathematics,  as  the  above  adduced 
inferences  illustrate,  provides  a  host  of  cases  in  which 
the  Applicative  principle  alone  is  explicitly  employed 
without  any  recourse  to  the  Implicative  principle.  These 
inferences  might  be  called  purely  Applicative'  in  con- 
trast to  the  syllogism,  which  in  our  analysis  has  been 
shown  to  involve  the  Implicative  as  well  as  the  Appli- 
cative principle.    Again  the  construction  of  the  logical 

*  Cf.  Chapter  I,  p.  11  and  onwards. 


^1 


%' 


V 

I 


calculus  involves  the  Implicative  as  well  as  the  Appli- 
cative principle,  and  will  be  discussed  later.  Before 
proceeding  to  this  topic,  we  must  complete  our  account 
of  mathematical  demonstration  by  an  analysis  of  mathe- 
matical  induction,  which  also  involves  both  principles. 
§  6.  Mathematical  induction  assumes  a  unique  place 
in  logical  theory.  It  resembles  other  forms  of  demon- 
strative induction,  which  will  be  discussed  in  a  later 
chapter,  where  it  will  be  shown  that  the  universal  mark 
of  this  type  of  induction  is  that  the  conclusion  demon- 
stratively inferred  asserts  for  every  case  what  has  been 
asserted  in  one  premiss  for  a  single  case.  The  possi- 
bility of  such  demonstration  rests  upon  the  logical 
character  of  the  other  premiss,  which  may  be  of  different 
types,  each  type  yielding  a  different  form  of  demonstra- 
tion. The  distinctive  characteristic  of  mathematical  in- 
duction is  that  it  is  concerned  with  finite  integers.  These 
constitute  a  discrete  series  beginning  with  the  integer  i, 
and  proceeding  step  by  step  in  the  construction  of  suc- 
cessive integers.  The  generation  of  each  successive 
integer  from  the  preceding  is  indicated  by  the  operation 
plus  I.  Thus,  using  the  illustrative  symbol  n  to  stand 
for  any  finite  integer,  the  operation  symbolised  as  ;« -f- 1 
will  yield  the  next  following  integer.  This  construction 
defines  the  general  conception  of  a  finite  integer  which 
is  fundamental  for  arithmetic.  The  method  of  mathe- 
matical induction  introduces  the  notion  of  function. 
Thus  f{n)  will  be  used  to  stand  for  any  proposition' 

^  The  functions  previously  adduced  were  mathematical,  i.e.  con- 
structs yielding  quantities,  whereas  the  function  here  introduced  is  pro- 
positional,  i.e.  a  construct  yielding  a  proposition.  And,  in  general,  the 
equating  of  two  mathematical  functions  yields  a  propositional  function. 


T 


134 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


135 


about  the  specific  integer  n,  where  variation  of  form 
will  be  represented  by  changing/ into  ^  say,  and  varia- 
tion of  reference  by  changing  n  into  m  say.  The  argu- 
ment in  its  general  form  will  consist  of  the  following 
assertions  of  two  premisses  and  of  the  inferred  con- 
clusion : 

Implicative  Premiss:  *The  proposition  f{n)  would 
imply  the  proposition  /(w+i)'  for  every  finite  in- 
teger n. 

Categorical  Premiss:  '/(i)'  holds. 

Conclusion:  Therefore  '/(«)'  holds  for  every  finite 
integer  n. 

In  this  argument  we  observe  that  the  conclusion  states 
categorically  what  is  stated  hypothetically  in  the  im- 
plicative premiss;  and  further  that  it  predicates  for 
every  case  what  is  predicated  for  a  single  case  in  the 
categorical  premiss.  Its  demonstrative  force  may  be 
shown  by  resolving  the  argument  into  a  succession  of 
steps.  Thus,  by  the  applicative  principle,  we  may  re- 
place in  the  implicative  premiss  «  by  i,  and  this  yields 
the  assertion  ' f{i)  would  implyy(2)';  then,  adding  the 
categorical  premiss  */(i),'  we  infer,  by  the  implicative 
principle,  */(2).*  Again,  replacing  n  by  2,  ^ f(2)  would 
imply  /(s),'  and,  adding  to  this  the  conclusion  of  the 
preceding  inference,  we  may  infer  ^/{zY  If  this  process 
is  indefinitely  continued  we  are  enabled,  by  use  merely 
of  the  applicative  and  implicative  principles,  to  infer 
successively y*(2),y*(3),y"(4),  etc.,  for  every  finite  integer. 
The  whole  argument  therefore  rests  merely  upon  the 
same  principles  as  are  involved  in  ordinary  deduction; 
and  yet  the  inference  is  of  the  nature  of  induction, 
because  the  conclusion  is  a  generalisation  of  the  same 


A 


?v^ 


r 


t 

% 


formula  that  the  categorical  premiss  lays  down  only  for 
a  single  case. 

The  following  is  a  simple  application  of  mathematical 
induction : 

Let  y(;^)  stand  for  the  proposition:  *The  sum  of 
the  first  n  odd  integers  =  «^'  We  have  first  to  establish 
the  implicative  premiss,  viz., 

'/{n)  would  implyy"(«4- 1).' 

Now/(«)  is  the  proposition 

*i+3  +  5  +  7  +  ...-l-(2«-i)  =  n\' 

and /(«-[- 1)  is  the  proposition 

*  1+ 3 -h  5  +  7  +  ...  +  (2;^ -  i)  +  (2/z  +  i)  =  (w  +  i)*/ 

Here  the  left  hand  side  of  the  equation/(«+  i)  is  ob- 
tained from  that  o(/{n)  by  adding  (2^+  i). 

Hence,  by  the  formula  for  the  square  of  the  sum  of 
two  numbers:  viz., 

(«-f  iy  =  ;^'-h(2«-M), 

the  conclusion  is  established  that 

*if/(«)  holds,  then/(^+  i)  would  hold.' 

Now/(i)  holds;  for  i  =  il  (Also/(2)  holds;  for 
i-h3  =  2':  and /(3)  holds;  for  i +3 -h 5  =  3'-) 

Hence,  having  established  the  implicative  premiss 
'/{n)  would  imply/(/^+  i),'  and  the  single  categorical 
premiss  '/(i),'  the  required  universal  '/{n)'  has  been 

proved. 

§7.  In  this  account  of  the  principles  employed  in 
establishing  general  algebraical  formulae,  special  em- 
phasis has  been  laid  on  the  novelty  of  the  conclusion 
as  compared  with  the  familiarity  and  obviousness  of  the 


136 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


137 


premisses  (including  the  axioms)  from  which  the  con- 
clusion is  drawn.  This  summary  account  of  the  methods 
and  results  of  deductive  reasoning  enables  us  to  meet 
what  has-^been  called  the  paradox  of  inference  in  a  more 
direct  way  than  that  explained  in  Chapter  I.  For  the 
existence  of  the  mathematical  calculus,  where  the  con- 
clusions are  absolutely  unknown  to  those  who  start  by 
admitting  as  self-evident  the  fundamental  premisses, 
constitutes  a  direct  refutation  of  the  arbitrary  dictum 
that  for  valid  inference  the  conclusion  must  not  contain 
more  than  what  is  already  known  in  asserting  the  pre- 
misses. 

The  notion  of  a  calculus  is  generally  associated  with 
elaborate  symbolism,  which  renders  possible  the  more 
complex  deductive  processes  in  logic  and  mathematics. 
As  a  question  of  history,  there  is  no  doubt  that  the  in- 
troduction of  such  simple  symbols  as  H- ,  — ,  x  ,  created 
a  revolution  in  mathematical  science,  and  rendered  it 
possible  to  make  advances  otherwise  unattainable. 
Again  it  is  an  equally  noteworthy  historical  fact  that 
the  best  formal  logicians,  such  as  Leibniz  and  Lambert, 
were  comparatively  unsuccessful  in  their  attempt  to 
develop  a  logical  calculus,  which  was  first  started  by 
Boole  on  lines  followed  by  all  subsequent  symbolists 
who  advanced  the  science.  Boole's  method  was  simply 
to  import  the  familiar  symbols  of  elementary  arithmetic 
into  logic,  making  use  of  the  fundamental  formulae  with 
which  algebraists  were  already  conversant.  In  this  way 
he  created  the  first  great  revolution  in  the  study  of 
formal  logic,  and  one  that  is  comparable  in  importance 
with  that  of  the  algebraical  symbolists  in  the  sixteenth 
century.    I  think,  however,  that  Boole's  procedure  has 


i 


< 


X 


■9       ^■: 


led  to  considerable  confusion  with  regard  to  the  relations 
between  the  logical  and  the  algebraical  calculus,  inas- 
much as  he  seems  to  have  supposed — in  common  with 
many  logicians  of  his  time— that  the  advance  achieved 
by  introducing  mathematical  formulae  into  logic  made 
logic  into  a  department  of  mathematics.  This  attitude 
of  Boole's  obstructed,  for  a  considerable  period,  the  in- 
vestigation of  the  foundations  of  mathematics,  which 
demanded  the  reversal  of  the  relationship  between  the 
two  sciences.  It  is  under  the  influence  mainly  of  Peano 
and  of  the  new  mathematicians  such  as  Cantor,  that  we 
now  recognise  mathematics  to  be  a  department  of  logic. 
The  current  phrase  mathematical  logic  is  ambiguous 
inasmuch  as  it  may  be  understood  to  mean  either  the 
logic  of  mathematics  or  the  mathematics  of  logic.  Now, 
in  my  view,  the  logic  or  rather  philosophy  of  mathe- 
matics is  a  study  which  ought  to  dispense  entirely  with 
symbolic  language.  It  must,  of  course,  explain  the  nature 
of  symbols  and  of  symbolic  methods,  and  account  for 
the  extraordinary  power  of  symbolism  in  deducing  with 
absolute  security  previously  unknown  formulae.  But 
the  philosophical  exposition  of  the  deductive  power  of 
mathematics  must  be  treated  in  language  the  under- 
standing of  which  requires  thought  of  a  profounder 
nature  than  that  required  in  merely  following  symbolic 
rules.  As  indicated  in  the  chapter  on  Symbolism  and 
Functions,  the  essential  purpose  of  symbolism  is  to 
economise  the  exercise  of  thought;  and  thus  symbolic 
methods  are  worse  thanuselessin  studyingthe  philosophy 
of  symbolism  or  of  mathematics  in  particular.  The  phrase 
*  mathematics  of  logic,'  on  the  other  hand,  merely  in- 
dicates a  certain  line  of  development  of  logic,  in  which 


138 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


139 


deductive  processes  are  reduced  to  strictly  demonstrative 
form  by  means  of  a  symbolism  founded  on  explicitly 
logical  axioms.  The  important  advances  in  this  direction 
have  been  systematised  with  extraordinary  success  in 
Whitehead  and  Russell's  great  work  Principia  Mathe- 
matical where  it  is  shown  how  pure  mathematics  can  be 
actually  developed  from  pure  logic.  The  value  of  the 
work  consists,  therefore,  in  reducing  mathematics  to 
logic,  and  not  at  all  in  reducing  logic  to  mathematics. 
I  shall  attempt  hardly  any  criticism  of  their  formal  de- 
velopment of  the  science,  and  shall  here  confine  myself 
to  the  principles  which  enter  into  its  very  elementary 
foundations. 

§8.  In  contrasting  the  mathematical  developments 
of  logic  with  the  ultimate  foundations  of  the  science,  it 
will  be  convenient  to  use  the  terms  premathematical 
and  mathematical  logic,  the  latter  of  which  introduces 
certain  novel  conceptions,  strictly  formal  in  character, 
in  addition  to  those  employed  in  the  former.  There 
are  certain  notions  common  to  the  premathematical  and 
mathematical  departments  of  logic,  and  of  these  we  have 
already  discussed  the  nature  of  functions,  illustrative 
and  short-hand  symbols,  variables,  brackets,  etc.,  which 
before  Peano  and  Russell  had  not  received  adequate 
recognition  in  logical  teaching;  they  apply,  however, 
over  a  wider  field  than  mere  mathematics,  and  must 
therefore  be  transferred  without  modification  from  the 
narrower  science  back  to  logic.  The  term  'formal'  as 
applied  to  these  conceptions  means  that  they  are  to  be 
understood  by  the  logician  as  such,  and  they  include, 
besides  those  primitive  ideas  which  are  to  be  understood 
without  definition,  also  derivative  ideas  which  are  com- 


*i 


pletely  defined  in  terms  of  primitive  ideas.  For  example, 
the  notions  of  implication,  alternation,  disjunction  and 
negation  are  formal,  and  of  these  we  may  take  negation 
and  alternation  as  understood  without  definition,  while 
the  others  can  be  defined  in  terms  of  these  two\  Aoain 
logical  categories  and  sub-categories  such  as  substantives 
|jroper,  primary  and  secondary  adjectives  and  proposi- 
tions, come  under  the  head  of  formal  conceptions.  There 
are  also  specific  adjectives  and  relations,  such  as  true, 
probable,  characterised  by,  comprised  in,  identical 
with,  which  are  formal;  and,  though  some  of  them  are 
ultimately  indefinable,  the  understanding  of  all  of  them 
is  essential  to  logical  analysis.  In  contrast  to  these 
formal  adjectives,  such  adjectives  as  red,  hard,  popular, 
virtuous y  etc.  are  termed  material,  because  their  meaning 
is  unessential  to  the  explication  of  logical  forms,  fn 
premathematical  logic  formulae  are  established  for  ail 
adjectives  as  such,  or  for  a  limited  set  of  adjectives 
comprised  in  such  a  sub-category  as  that  of  secondary 
adjective.  The  range  over  which  these  formulae  hold 
must  be  said  to  be  material,  though  it  necessarily  com- 
prises adjectives  which  may  happen  to  be  formal,  i.e.  to 
have  specifically  logical  significance.  Passing  from  pre- 
mathematical to  mathematical  logic,  we  find  that  new 
specific  adjectives,  having  essentially  logical  significance 
and  coming  therefore  under  the  head  of  formal  concep- 
tions, are  introduced.  We  may  specially  mention  in- 
tegers and   ratios.     Integer   is   a  logical   sub- category 

^  This,  at  any  rate,  is  the  procedure  of  the  Principia  Mathematica', 
but,  while  undoubtedly  permissible  from  the  point  of  view  of  the 
logical  calculus,  it  is  open  to  serious  philosophical  criticism,  which  I 
have  given  elsewhere. 


140 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


141 


comprised  in  the  general  Q.2X^goxy  adjective ;  and  ratio  is  a 
logical  sub-category  comprised  in  the  general  category 
relation ;  but  what  constitutes  the  new  feature  in  mathe- 
matical logic  is  that  each  specific  integer  and  each 
specific  ratio  has  itself  essentially  logical  significance, 
while  at  the  same  time  formulae  hold  for  all  integers 
and  again  for  all  ratios.  Premathematical  logic  on  the 
other  hand  can  only  establish  formulae  holding  for  ad- 
jectives in  general  or  for  secondary  adjectives  in  general. 
This  distinction  carries  with  it  the  further  result  that 
premathematical  logic  can  only  use  illustrative  adjectival 
symbols  as  variables  over  a  range  of  variation  covering 
the  whole  category  adjective,  or  the  whole  sub-category 
secondary  adjective)  while  in  mathematical  logic  there 
occur  illustrative  symbols  for  variables  covering  the 
range,  in  the  one  case  integer,  in  the  other  ratio.  Con- 
sider for  example  such  an  illustrative  symbol  as  m  in 
ordinary  or  premathematical  logic.  The  specific  values 
that  can  be  substituted  for  this  variable  are  material; 
for  the  formal  character  of  such  of  them  as  have  speci- 
fically logical  significance  is  irrelevant  to  the  truth  of 
the  formulae.  In  mathematical  logic,  on  the  other  hand, 
all  the  specific  values  which  can  be  substituted  for  a 
symbol  m  standing  for  any  integer,  say,  or  a  symbol  / 
standing  for  any  ratio,  denote  formal  conceptions.  Again 
it  is  obvious  that,  besides  the  formulae  which  hold  for 
adjectives  in  general,  there  are  innumerable  additional 
formulae  holding  for  integers  or  for  ratios;  and  this 
accounts  for  the  variety  and  complexity  of  mathematics 
as  compared  with  premathematical  logic.  But  the  es- 
sential distinction  between  the  two  sciences — or  rather 
the  two  departments  of  logical  science — lies  in  the  point 


«   4 


V 


already  urged,  namely  that  every  specific  adjective  within 
a  certain  mathematical  range  has  itself  a  logically  de- 
termined value ;  whereas  no  logically  determined  value 
can  be  assigned  to  adjectives  in  general  which  enter 
into  premathematical  logic.  This  distinction  may  be 
summed  up  in  other  words  by  taking  the  two  antitheses 
material  and  formal,  and  constant  and  variable,  which 
combined  give  the  four  cases  formal  variables,  foniial 
constants,  material  variables  and  material  constants. 
Now  premathematical  logic  uses  formal  constants  and 
material  variables  (and  also  in  Mr  Russell's  work  material 
constants),  but  nowhere  formal  variables.  On  the  other 
hand  mathematics  uses  formal  constants,  material  vari- 
ables, and  also  formal  variables.  It  is  therefore  the  use 
of  formal  variables  that  fundamentally  distinguishes 
mathematics  from  premathematical  logic. 

§  9.  To  continue  our  account  of  the  relation  between 
the  premathematical  and  mathematical  departments  of 
logic,  we  must  next  define  and  illustrate  the  nature  of 
those  formal  elements  which  are  never  expressed  by 
variable  symbols,  and  therefore  come  under  the  head  of 
formal  constants.  To  these,  the  name  connecttves  will 
be  given.  The  first  division  under  this  head  includes 
what  are  known  as  operators  in  mathematics,  such  as 
plus,  minus,  multiplied  by,  divided  by,  as  well  as  ana- 
logous logical  operators  such  as  and.  or\  Jioi,  if.  Thus 
the  operation  'm-^n'  where  m,  n  stand  for  determinate 
numbers,  yields  a  certain  determinate  number;  and 
analogously  the  operation  /  and  q^  where  /,  q  stand 
for  determinate  adjectives,  yields  a  certain  determinate 
adjective.  This  is  most  clearly  seen  when  a  proper 
name  has  been  invented  to  stand  for  the  comoound 


142 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


143 


construct  as  well  as  for  each  of  the  constituents  them- 
selves: thus,  the  operation  *  three-plus-five'  yields  the 
number  *  eight';  the  operation  'rational-and-animated' 
yields  the  adjective  'human.'  The  analogy  goes  one 
step  further  when,  in  place  of  the  simple  predication 
yields,  we  use  the  complex  yields-what-is-yielded-by : 
thus,  the  operation  'm  plus  n  yields-what-is-yielded-by 
the  operation  *;^  plus  m\  the  operation  'p  and  q  yields- 
what-is-yielded-by  the  operation  'q  and/.'  Now  neither 
in  logic  nor  in  mathematics  is  it  ever  required  to  use 
illustrative  or  variable  symbols  to  stand  for  formal 
operators  like  plus  or  a;/^— the  reason  being  that  no 
formula  which  holds  for  one  operator  will  hold  if  we 
substitute  indiscriminately  any  other  operator.  Hence, 
if  symbols  are  used  for  formal  operators,  these  come 
under  the  head  of  short-hand  symbols,  and  never  under 
the  head  of  illustrative  or  variable  symbols.  Thus  the 
operators  both  of  logic  and  of  mathematics  enter  as 
formal  constants,  never  as  variables. 

In  the  second  division  of  connectives  are  to  be  in- 
cluded certain  relational  predications  which  must  be 
systematically  illustrated  and  classified  according  to 
their  different  properties.  Of  these,  the  five  of  most 
fundamental  importance  are  the  relational  predications : 
identical  with,  implied  by,  characterised  by,  comprised 
in,  included  in,  together  with  their  cognates.  These 
are  formal,  and  to  represent  them  I  shall  introduce  the 
short-hand  symbols:  X,l\  \,\\  %  x\  k,  k\  v,v  respec- 
tively. These  five  formal  connectives  are  absolutely 
distinct  from  one  another,  although  they  have  been 
frequently  confused  by  logicians ;  and  this  distinctive- 
ness is  sufficient  to  account  for  the  fact  that  they  are 


«   4 


■ 


never  represented  by  variable  symbols  for  which  one 
could  replace  another.  Thus:  in  the  predication  xiy, 
the  symbols  x  and  y  must  stand  for  entities  belonging 
to  one  and  the  same  assigned  category;  but,  in  the 
predication  XKy,  x  and  jj/  must  stand  respectively  for  an 
item  or  member  and  an  enumeration  or  class;  and,  in 
the  predication  xxy,  x  and  y  must  stand  respectively 
for  a  substantive  and  an  adjective.  Again,  while  I  con- 
nects entities  belonging  to  any  the  same  category,  X 
connects  only  propositions  or  adjectives  or  relations; 
and  V  connects  only  classes  or  enumerations  of  the  same 
order.  And  yet  again :  the  relation  identity  is  reflexive, 
symmetrical,  and  transitive;  but  the  relations  ckarac- 
terised  dy  Sind  comprised  in  are  a-reflexive,  a-symmetrical, 
a-transitive;  while  the  relations  implying  and  included 
in  are  reflexive  and  transitive  but  neither  symmetrical 
nor  a-symmetrical.  The  five  connectives  above  enu- 
merated may  be  said  to  be  on  the  borderland  between 
premathematical  and  mathematical  logic.  There  are, 
however,  many  formal  connectives  which  belong  ex- 
clusively to  mathematics,  of  which  the  most  funda- 
mental is  equals  universally  represented  by  the  short- 
hand symbol  = .  There  is  serious  danger  of  confusing 
equal-to  with  identical-wit k  because  they  agree  in  pos- 
sessing the  properties  reflexive,  symmetrical  and  tran- 
sitive (to  the  consideration  of  which  we  shall  have  to 
return  later).  Other  important  connectives  in  logic  and 
algebra  are  derivative  from  those  above  enumerated  as 
fundamental.  Classifying  fundamentals  and  derivatives 
according  to  their  properties  we  have  the  following 
table,  where  the  initials  F,  S,  T  stand  respectively  for 
reflexive,  symmetrical  and  transitive,  and  the  suffix  a 


144 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


145 


means  *for  all  cases,'  e  *for  no  cases/  and  oi  *for  some 
but  not  all  cases.' 


Formal  Relations 

Identical,  equal,  co-implicant,  coincident  

Differing  by  unity,  co-opponent,  co-remainder 

Greater,  less,  sub-  and  super-implicant,  sub-  and  super- 
incident     •.»         •••         •••        _•••         •••         •••        ••• 

Sub-  and  super-opponent,  sub-  and  super-remainder... 

Not  greater,  not  less,  implying,  implied  by,  included  in, 
including  ...         ...         ...         ... 

Other  than,  unequal,  disjunct,  alternant,  co-exclusive, 
co-exhaustive 

Not  disjunct,  not  alternant,  not  co-exclusive,  not  co- 
exhaustive... 

Not  implying,  not  implied  by,  not  included  in,  not 
including  ... 

Comprising,  comprised  in,  characterising,  charac- 
terised by ... 


Properties 

Fa  Sa  ^a 

Ft  ^a  ^e 

Ft,  *S'g  7a 

F  9  T  ■ 

/'a  ^o»  -'  o 

/*«  ^a.  -^  oi 

Fa  Sa  Toi 

-*  6  '^oi  -^  oi 

Fp.  Sg  Tg . 


e      "^e 


§  10.  Not  only  formal  constants  but  also  material 
variables  enter  in  the  same  way  into  mathematics  as  into 
premathematical  logic.  The  particular  values  which  any 
material  variable  may  assume  are  unessential  for  pure 
logic  and  pure  mathematics;  and  enter  as  significant 
factors  only  into  applied  logic  or  applied  mathematics. 
For  example,  a  variable  representing  any  substantive 
or  any  adjective  is  replaced  by  a  particular  substantive 
or  a  particular  adjective  only  when  the  general  formulae 
established  by  logic  are  applied  to  concrete  propositions. 
Similarly  the  purely  formal  notion  of  magnitude  or  of 
quantity,  which  enters  into  mathematics,  is  applied  to 
several  different  species  and  sub-species  such  as  mass, 
volume,  intensities  of  different  kinds,  etc.,  the  dif- 
ferentiae of  which,  not  being  expressible  in  terms  of 
pure  mathematical  conceptions,  must  be  determined 
materially.    Thus,   for  instance,   in  the  mathematical 


-/ 


formula  3?'  +  5i^  =  8i^»  i^  enters  as  a  material  variable 
standing  for  any  quantity;  and  3.  5»  8,  = ,  -f ,  as  also  the 
category  quantity  itself,  enter  as  formal  consta n ts.  B  u t  i  n 
applying  the  material  variable  q  to  deduce  the  equation 

3  feet  +  5  feet  =  8  feet,  or  3  ohms  ^  5  ohms  =  8  ohms, 

the  terms  foot,  ohm,  as  species  of  the  genus  quantity, 
have  to  be  defined  by  means  of  conceptions  outside 
the  range  of  pure  mathematics.  In  this  wav  we  see 
that  variable  symbols— material  as  regards  their  range 
of  application — entering  into  premathematical  and 
mathematical  logic,  assume  their  particular  values  when 
logical  theorems  are  applied  to  experimental  matter. 
Having  shown  then,  as  regards  both  formal  constants 
and  material  variables,  that  general  logic  agrees  in  cill 
respects  with  mathematics,  the  conclusion  follows  that 
the  latter  fundamentally  differs  from  the  former  in  the 
sole  fact  that  it  introduces  formal  variables. 

§  II.  Before  examining  the  characteristics  of  the 
specifically  mathematical  notion  *  equals*  iinon  which 
its  symmetry  and  transitiveness  depend,  we  will  con- 
sider the  wider  problem  of  relations  in  general  possess- 
ing these  two  properties.  There  is  one  mode  of  con- 
structing such  relations  which  has  very  wide  application 
and  is  of  great  importance  in  logical  theory,  viz. 

^x  is  r  to  the  thing  that  is  r  to  z' 

Here  the  word  the  indicates  that  r  is  a  man\  one  rela- 
tion. 1  shall  call  *  the  thing'  to  which  reference  is  made 
in  the  above  formula  the  intermediary  term,  and  the 
relation  r  the  generating  relation.  Thus,  i^iven  an  in- 
termediary term  and  a  many-one  generating  relation, 
we  can  always  construct  by  (what  is   called)   relative 


J.  L.  II 


10 


146 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


147 


multiplication  a  derived  relation  which  is  symmetrical 
and  transitive.  Representing  the  intermediary  by  the 
symbol  y,  the  relation  of  :r  to  -^  may  be  otherwise  ex- 
pressed  by  the  conjunctive  proposition : 

'x  is  f  to  ^  and  ^  is  ?  to  yl 
where  it  is  to  be  understood  that  there  is  some  uniquely 
determined  entity  (sayj>/>to  which  ^  and  ^  stand  in  the 
relation  r\  i.e.  f  is  a  many-one  relation. 

Now  the  theorem  that  any  relation  so  constructed 
is  symmetrical  and  transitive  requires  no  discussion 
and  is  universally  admitted  ;  but  the  converse  theorem 
—that  any  symmetrical  and  transitive  relation  can  be 
exhibited   by  this  mode  of  construction— cannot  be 
assumed  to  be  true  without  careful  examination.    To 
this  converse  theorem  Mr  Russell  gives  the  name  *the 
principle  of  abstraction';  and  professes  to  have  proved 
its  truth  by  a    process   involving  highly  complicated 
symbolism.    It  is  quite  easy,  however,  to  explain  the 
nature  of  his  proof  without  recourse  to  such  symbolism. 
Thus,  let  i  be  a  symmetrical  and  transitive  relation  ; 
then,  in  order  to  prove  the  theorem,  we  have  to  dis- 
cover an  intermediary  entity  and  a  generating  relation 
in  terms  of  which  /  may  be  constructed.    The  inter- 
mediary entity  for  the  relational  predication  'xx^tloz 
is,  in  Mr  Russell's  proof,  '  the  class  of  things  comprismg 
X  together  with  everything  such  as  z  for  which  ''x  is  t  to 
z'  holds.'  The  required  generating  relation  r  is  the  rela- 
tion o(  being  comprised  in  \  hence  the  proposition  'x  is 
ttoz'  is  resolved  into  the  form: 

X  is  comprised  in  the  class 
(defined  as  comprising  everything  to  which  x  is  /) 

which  comprises  z. 


k 


Here  the  intermediary  entity  is  a  class  uniquely  defined 
in  terms  of  ^  and  /,  and  therefore  the  relation  in  which 
X  or  any  other  item  stands  to  the  intermediary  is  a 
many-one  relation.  Now  what  Mr  Russell  has  suc- 
ceeded in  proving  in  this  way  is  proved  with  absolutely 
demonstrative  validity;  but  my  first  comment  is:  has 
he  proved  what  he  undertook  to  prove  ?  In  one  sense 
he  has  proved  too  much,  and  in  another  sense  he  has 
proved  nothing  whatever  that  is  relevant.  He  has 
proved  too  much  in  the  sense  that  he  has  discovered  an 
intermediary  entity  which  would,  mutatis  mutandis, 
apply  to  every  possible  symmetrical  and  transitive  rela- 
tion, such  as  contemporaneous,  compatriot,  co-implicant, 
co-incident,  as  well  as  equal.  Thus  he  has  proved  that, 
for  the  resolution  of  the  relation  equals,  we  must  take  as 
intermediary  '  the  class  of  quantities  equal  to  any  given 
quantity';  for  the  relation  contemporaneous y  'the  class 
of  events  contemporaneous  with  any  given  event'; 
for  the  relation  compatriot,  'the  class  of  persons  that 
are  compatriots  of  any  given  person  ' ;  and  so  on.  But 
what  he  set  out  to  discover  as  the  required  inter- 
mediary was,  in  the  case  of  equality,  a  certain  magni- 
tude', in  the  case  of  compatriot,  a  certain  country,  in 
the  case  of  contemporaneous,  a  certain  date;  and  so 
on.  He  has  not  proved  that  there  is  a  certain  magni- 
tude that  all  equal  quantities  possess ;  nor  a  certain 
country  to  which  all  compatriots  belong;  nor  a  certain 
date  to  which  all  contemporaneous  events  are  to  be 
referred.  Moreover,  in  taking  as  his  intermediary  a 
certain  uniquely  determined  class,  it  seems  obvious  that 
Mr  Russell's  alleged  proof  is  incomplete,  unless  we  can 
assert  that  there  are  such  entities  as  classes,  and  thti 


10 — 2 


148  CHAPTER  VI  i 

validity  of  this  assertion  is  explicitly  denied  by  him  :  ot 
rather  he  holds  that  there  is  no  necessity  in  the  deduc- 
tions of  logic  and  mathematics  to  assume  that  there  are 
classes,  although  without  this  assumption  his  proof  of 
the  principle  of  abstraction  completely  breaks  down. 

I  do  not,  however,  wish  to  press  my  criticism  of 
Mr  Russqll  further,  but  rather  to  expound  what  ap- 
pears to  me  to  be  the  true  view  on  the  nature  of 
abstraction.  The  cases  in  which  the  principle  comes 
into  consideration  may  be  distinguished  according  as 
the  intermediary  is  of  the  nature  of  a  substantive  such 
as  country,  or  of  the  nature  of  an  adjective  such  as 
magnitude.  In  applying  the  attempted  proof  of  the 
principle  of  abstraction  to  such  a  relation  as  compatriot, 
Mr  Russell  argues  as  if  we  knew  this  relation  to 
be  symmetrical  and  transitive  independently  of  our 
knowledge  that  a  person  can  belong-to  (r)  only  one 
country  (jv);  whereas  it  is  obvious  that  we  have  con- 
structed the  derivative  relation  compatriot  by  means  of 
the  prior  notions  country  and  belonging-to.  Hence,  no 
such  case  as  compatriot  can  be  used  to  prove  the  prin- 
ciple of  abstraction,  but  only  to  illustrate  the  theorem 
of  which  the  principle  of  abstraction  is  the  converse. 
Where  the  intermediary  is  adjectival,  e.g.  colour,  pitch, 
magnitude,  the  principle  directly  raises  the  issue  of  the 
connection  and  distinction  between  a  determining  ad- 
jective and  the  class  that  it  determines.  In  the  case  of 
an  adjectival  intermediary,  our  general  formula 

'x  is  r  to  the  term  (say  jj/)  that  is  r  to  z' 

must  be  expressed  in  a  special  form  in  which  the  gene- 
rating relation  i^r)  is  to  stand  for  characterised-by  (x), 


FUNCTIONAL  DEDUCTION 


149 


'♦ 


4. 


and  the  intermediary  term  i^y)  is  to  stand  for  a  specific 
determinate  under  a  specific  determinable,  thus : 

'x  is  characterised-by  the  determinate  adjective 

that  characterises  z' 

Here  the  uniqueness  of  the  intermediary  term  is  secured 
by  the  disjunctive  principle  of  adjectival  determination 
expressed  (Part  I,  Chapter  XIV)  in  the  form:  'Nothing 
can  be  characterised  by  more  than  one  determinate 
under  any  assigned  determinable.'  Now,  since  any  one 
substantive  may  be  characterised  under  many  different 
determinates,  the  intermediary  term  must  specify  the 
determinable,  or  ground  of  comparison,  upon  which  the 
symmetry  and  transitiveness  of  the  derived  relation 
depend.    Thus, 

'x  is  characterised  by  the  colour  that  characterises  z' 
or  'x  is  characterised  by  the  shape  that  characterises^,' 
or  ^x  is  characterised  by  the  size  that  characterises  z! 

Any  of  these  three  propositions  may  be  significantly 
asserted  of  the  same  subjects  x  and  z,  if  these  are 
bounded  surfaces  distinguished  from  one  another  by 
determinate  localisation;  and  the  relation  of  x  to  z  thus 
constructed  is  transitive  (as  well  as  symmetrical)  pro- 
vided that  the  colour,  shape  or  size  is  strictly  deter- 
minate. With  this  proviso,  we  may  say  that  x  and  z 
are  equivalently  coloured,  equivalently  shaped  or  equi- 
valently  sized,  as  the  case  may  be.  Such  symmetrical 
and  transitive  relations  between  the  substantives  x  and 
z  must  be  distinguished  from  the  symmetrical  and  tran- 
sitive relsitionidentity  which  holds  between  the  adjectives 
described  as  the  colour  of  :tr  and  the  colour  of -s:,  the  shape 
ofx  and  the  shape  of  z,  or  the  size  of  x  and  the  size  of  z. 


150 


CHAPTER  VI 


FUNCTIONAL  DEDUCTION 


151 


Now  magnitude — like  any  other  adjectival  determinable 
— must  first  be  abstracted  as  a  character  in  order  that  by 
Its  means  we  can  construct  the  class  of  equally  sized 
objects.  Thus  it  is  just  as  absurd  to  define  the  size  of 
X  in  terms  of  '  the  class  of  objects  that  are  equal  in  size 
to  ;r '  as  to  define  the  colour  of  x  in  terms  of  *  the  class 
of  objects  that  are  equivalent  in  colour  to  x! 

§  12.  To  secure  that  the  relations  constructed  by 
means  of  the  above  formula  shall  be  symmetrical  and 
transitive,  it  is  necessary  to  specify,  not  only  such  differ- 
ences as  those  between  colour,  shape,  etc.,  but  also 
differences  within  the  general  notion  magnitude,  con- 
stituting various  kinds  or  species  of  magnitude.  For 
just  as  colours  and  sounds  are  incomparable  with  one 
another,  since  they  must  be  characterised  under  dif- 
ferent determinates,  so  there  are  distinct  determinates 
subsumable  under  the  superdeterminable  magnitude. 
Taking  some  of  Mr  Russell's  suggestive  examples,  we 
note  that  the  magnitude  of  pleasure  predicable  of  an 
experience  is  incomparable  with  the  magnitude  of  area 
predicable  of  a  surface,  and  that  these  again  are  in- 
comparable  with  the  magnitude  of  duration  predicable 
of  an  event.  Hence  pleasure-magnitude,  area-mag- 
nitude, duration-magnitude,  are  three  distinct  deter- 
minates, predicable  only  of  experiences,  surfaces,  and 
events  respectively.  In  ordinary  usage  the  word 
magnitude  is  omitted  when  reference  is  made  to  the 
determinates  in  question  ;  but  in  specifying  the  *area' 
of  a  surface,  we  are  in  point  of  fact  specifying  a  kind 
of  magnitude  ;  so  in  specifying  the  '  duration '  of  an 
event  we  are  specifying  another  kind  of  magnitude; 
and  in  specifying  the  'pleasure'  of  an  experience,  we 
are  specifying  yet  another  kind  of  magnitude.    The 


analogy  here  drawn  between  area  or  duration  on  the 
one  hand,  and  pleasure  on  the  other  will  probably  be 
disputed  because  pleasure  is  so  often  used  in  its  concrete 
sense  to  mean  *  pleasurable  experience '  as  well  as  in  its 
abstract  sense  to  mean  '  the  pleasure  of  a  (pleasurable) 
experience.'    Now  it  happens  that  a  pleasurable  ex- 
perience may  be  characterised  under  at  least  two  dif- 
ferent determinates  of  magnitude;  viz.  pleasure-mag- 
nitude  and    duration-magnitude,   the  latter  of  which 
applies  in  the  same  sense  to  any  event  whatever  that 
may  last  through  a  period  of  time.   Here  it  is  important 
to  note  that  pleasure-magnitude  and  duration-magni- 
tude, etc.  are  not  determinates  under  the  one  deter- 
minable magnitude,   but  different  species  included  in 
the  genus  magnitude.     They  may  therefore  be  con- 
veniently   termed     sub-determinables    of    magnitude, 
each  generating  its  own  determinates,  which  are  in- 
comparable with  the  determinates  generated  by  any 
other.     Thus  magnitude  does  not  generate   its   sub- 
determinables  in  the  way  in  which  a  determinable  gene- 
rates its  determinates.    An  experience,  a  surface,  an 
event  are  substantives  belonging  to  different  categories 
of  which  pleasure,  area,  or  duration  may  be  respec- 
tively predicated  as  adjectives  ;  but  a  specific  pleasure- 
magnitude,  or  area-magnitude,  or  duration-magnitude 
is  related  to  its  respective  species  of  magnitude  as  a 
determinate  to  its  determinable.    We  shall  proceed  in 
the  next  chapter  to  examine  and  classify  the  fundamental 
kinds  of  magnitude,  to  which  reference  is  here  made. 

§  13.  It  remains  to  point  out  one  highly  important 
characteristic  which  distinguishes  pure  or  pre-mathe- 
matical  logic  from  mathematics  proper.  I  n  both  branches, 
the  two  principles  of  inference  termed  Applicative  and 


\ 


i 


152 


CHAPTER  VI 


Implicative  are  employed  in  the  procedure  of  functional 
inference,  and  these  alone.  But  the  peculiarity  of 
pre-mathematical  deduction  is  that  it  lays  down  two 
formulae  of  imp  lication{px\\^x  2.^  primitive  or  as  derived) 
which  are  virtually  equivalent  respectively  to  the  Appli- 
cative and  Implicative  Principles  themselves.  The 
formulae  in  question  may  be  thus  expressed : 

(i)  Applicative  formula-.  Any  predication  that  holds 
for  every  case  x  would  formally  imply  that  the  same 
predication  holds  for  a  given  case  a, 

(2)  Implicative  formula :  For  any  case  Xy  y,  the  com- 
pound '  ''x"  and  ''x  would  imply  y' '  would  formally 
imply  'y! 

We  must,  therefore,  explain  the  distinction  between 
Principles  of  Inference,  on  the  one  hand,  and  Formulae 
of  Implication,  on  the  other  hand.  In  all  formulae  of 
implication,  the  implicans  and  implicate  stand  indif- 
ferently for  propositions  that  are  to  be  materially  or 
formally  certified.  But,  when  a  formula  of  implication 
is  used  as  a  premiss  in  the  process  of  deduction,  its 
implicans  must  first  be  formally  certified  in  order  that 
its  implicate  may  be  formally  certified.  This  inference 
is  made  by  a  direct  application  of  the  implicative  prin- 
ciple. And  again,  every  formula  of  implication  holds 
for  all  cases  coming  under  an  assigned  form  ;  hence  the 
inferences  from  any  formula  of  implication  are  made  by 
a  direct  application  of  the  applicative  principle.  The  fact 
that  every  step  by  which  we  advance  in  the  building 
up  of  the  logical  calculus  requires  both  the  Applicative 
and  the  Implicative  principles  of  inference,  and  these 
alone,  establishes  their  sovereignty  over  all  deductive 
processes. 


\' 


]) 


CHAPTER  VII 

THE  DIFFERENT  KINDS  OF  MAGNITUDE 

§  I.  The  term  magnitude^  as  is  suggested  by  its 
etymology,  denotes  anything  of  which  the  relations 
greater  or  less  can  be  predicated;  and  it  is  only  if  M 
and  N  (say)  are  magnitudes  of  the  same  kind  that  M 
can  be  said  to  be  greater  or  less  than  N.  I  have  taken 
magnitude  to  be  an  adjectival  determinable,  or  rather  a 
class  of  adjectival  determinables  including  several  dis- 
tinct kinds.  That  of  which  a  determinate  magnitude 
of  a  specific  kind  may  be  predicated  stands,  relatively 
to  its  magnitude,  as  substantive  to  adjective;  but  it  may 
be  either  an  existent,  i.e.  substantive  proper  (in  which 
case  the  magnitude  predicated  is  a  primary  adjective) 
or  itself  an  adjective  (in  which  case  the  magnitude  pre- 
dicated is  a  secondary  adjective).  In  order  to  keep 
clear  the  distinction  between  the  adjectives  of  magni- 
tude themselves  and  the  substantives  of  which  magni- 
tude is  predicable,  a  separate  terminology  ought  strictly 
to  be  applied  to  the  latter.  A  striking  case  where 
language  supplies  us  with  the  logically  required  termi- 
nological distinction  is  that  of  'longer'  and  'shorter* 
predicated  of  lines — to  the  lengths  of  which  the  terms 
'greater'  and  'less '  are  applied.  It  would  be  convenient, 
for  the  purposes  of  a  general  exposition  of  magnitude, 
to  restrict  the  application  of  the  terms  'greater'  and 
'less'  to  magnitudes,  and  to  adopt  the  corresponding 
terms    *larger'    and    'smaller'  for  that  of  which   the 


154 


CHAPTER  VII 


magnitudes  are  predicated.  For  example:  the  class 
compositae  is  larger  or  smaller  than  the  class  violaceaCy 
according  as  the  number  of  compositae  is  greater  or  less 
than  the  number  of  violaceae^;  the  period  1815  to  1832 
may  be  called  larger  than  the  period  1714  to  1720, 
inasmuch  as  the  temporal  magnitude  of  the  former  is 
greater  than  that  of  the  latter.  Now  for  every  distinct 
kind  of  magnitude  there  is  a  corresponding  distinct  kind 
or  category  of  entity  of  which  it  can  be  predicated ;  and 
hence,  though  it  is  strictly  illogical,  yet  it  is  legitimate 
and  usual  to  apply  the  same  terms,  such  as  extensive 
and  intensive,  to  distinguish  both  between  the  different 
kinds  of  magnitude  and  between  the  corresponding 
different  kinds  of  entities  which  bear  to  the  magnitude 
the  relation  of  substantive  to  adjective.  From  these 
preliminary  remarks,  we  may  pass  to  an  examination  of 
the  nature  of  different  kinds  of  magnitude,  beginning 
with  number,  which  is  the  most  fundamental  of  all. 

§  2.  Integral  number  is  an  adjective  exclusively 
predicable  of  what  we  call  classes,  including  enumera- 
tions; two  classes  being  said  to  be  numerically  equal 
when  the  number  predicable  of  the  one  is  identical  with 
that  predicable  of  the  other.  I  think  it  is  legitimate  to 
maintain  that  the  two  notions  class  and  number  are  not 
independently  definable,  but  each  definable  only  in  its 
relation,  the  one  as  the  only  appropriate  substantive  for 

^  This  may  mean  either  that  the  number  of  existing  plants  com- 
prised in  the  genus  is  greater  or  less,  or  that  the  number  of  infimae 
species  included  in  the  genus  is  greater  or  less.  It  is  obvious  that  these 
two  modes  of  determining  numerical  comparison  do  not  necessarily 
tally.  It  will  be  shown  later  that  the  same  distinction  holds  as  regards 
the  number  oi points  in  a  line  and  the  number  of  linear /^r/'j  (equal 
or  unequal)  into  which  it  may  be  exhaustively  and  exclusively  divided. 


t' 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        155 

the  other  as  its  only  appropriate  adjective.  The  common 
habit  of  representing  classes  by  closed  figures  may  lead 
to  the  false  supposition  that  the  members  of  a  class 
can  as  such  be  arranged  in  some  kind  of  proximity  to 
one  another  within  an  enclosed  space.    But  when  the 
items  to  be  comprised  in  a  class  have  relations  meta- 
phorically called  near  or  far,  they  constitute  not  merely 
a  class  but  a  series  or  ordered  class.    Now  in  modern 
mathematics   the    appropriate    number-adjective  of  a 
class  conceived  independently  of  any  arrangement  or 
order  of  its  items,   is  known  as  a  cardinal  number; 
whereas  of  a  series  or  ordered  aggregate  the  appropriate 
numerical  adjective  is  known  as  an  ordinal  number. 
When  a  class  or  enumeration  comprises  a  finite  number 
of  items,  then,  in  whatever  order  the  items  may  be 
enumerated,  we  reach  the  same  ordinal  number,  and 
this  number  agrees  with  the  cardinal  number;  but  for 
transfinite  aggregates,  which  have  been  introduced  into 
modern  arithmetic,  this  agreement  no  longer  holds; 
and  consequently  the  fundamental  distinction  between 
ordinal  and  cardinal  numbers  is  required.    Readers  are 
referred  particularly  to  Mr  Russell's  Principles  of  Mathe- 
matics for  the  full  development  of  this  topic,  which  is 
outside  the  compass  of  my  work. 

§  3.  The  psychological  aspect  of  number  is  revealed 
by  analysing  the  process  of  counting.  In  this  process 
we  establish  numerical  equality  between  a  set  of  things, 
on  the  one  hand,  and  a  set  of  number-names  temporarily 
attached  to  the  things,  on  the  other  hand.  Hence 
counting  is  a  special,  and,  in  some  respects,  a  unique 
case  of  correlation  between  the  things  upon  which  names 
are  imposed  and  the  names  that  are  imposed  upon  the 


,'l 


156 


CHAPTER  VII 


things.  Ideally  language  requires  that  any  given  proper 
name  should  denominate  one  and  only  one  thing,  and 
conversely  that  any  given  thing  should  be  denominated 
by  one  and  only  one  proper  name ;  or  briefly,  that  there 
should  be  a  one-one  correlation  between  the  names  of 
things  and  the  things  named.  If  this  relation  held,  it 
wciuld  follow  that  the  class  of  names  would  be  numeri- 
cally equal  to  the  class  of  things  named.  Actually, 
however,  this  ideal  is  not  realised;  for  the  same  thing 
often  has  many  names,  and  the  same  name  is  often 
attached  to  many  things.  It  is  worth  pointing  out  that 
there  may  still  be  numerical  equality  in  spite  of  there 
not  being  a  one-one  correlation  between  names  and 
things  named.  For  example :  let  Ry  Qy  M,  T,  £/  be  a 
set  of  names,  and  /c,  cr,  ^,  x»  ^  a  set  of  things  named. 
Then  suppose  that 

R  names  fc  or  o-;  Q  names  k  or  o-;  M  names  a-  or 
d  ov  x\  7"names  ^ ;  and  U names  ^ or  <^  or  /c  or  ;( ; 

so  that 

K  is  named  R  or  Q  or  U\  c  is  named  R  or  Q  or 
M\  6  is  named  M or  U\  x  ^^  reamed  M or  Tor 
U\  and  <^  is  named  U, 

Here  the  denominating  correlation  is  not  one-one  but 
many-many,  and  yet  the  names  and  the  things  happen 
to  be  numerically  equal.  How  then  do  we  establish  the 
fact  that  the  number  of  items  in  the  enumeration  R,  Q, 
M,  Ty  U  is  the  same  as  that  in  the  enumeration  /c,  cr, 
0y  X>  4^^  What  we  do,  where  there  is  no y^^/^^^/ correla- 
tion, is  to  institute  what  I  shall  call  a  factitious  corre- 
lation ;  by  which  I  mean  one  which  is  not  inherent  or 
objective,  but  arbitrarily  imposed  by  the  counter.  In  the 
adduced  instance — in  order  to  establish  the  numerical 


{■ 


i 


'f 


THE  DIFFERENT  KINDS  OF  MAGNITUDE       157 

equality  between  the  enumerations  R,  Q,  M,  T,  U, 
and  AC,  o",  ^,  x»  ^ — we  must  mentally  attach,  either  in 
thought  or  in  figurative  imagery,  R  say  to  6,  Q  to  <^, 
7"to  cr,  U  \.o  K,  M  \.o  yj,  where  the  items  of  the  two 
sets  have  been  indiscriminately  permuted  and  attached. 
We  can  now  analyse  the  mental  act  of  counting  as  a 
special  case  of  factitious  correlation.  The  essential 
psychological  requisite  is  that  we  should  learn  to  cmi- 
merate  a  set  of  arbitrary  names  in  a  fixed  or  invariable 
temporal  order  from  the  first  onwards ;  and  these  names 
are  attached  temporarily  to  the  objects  to  be  counted, 
in  this  respect  differing  from  names  in  general  which 
have  fixed  denotation.  For  example :  let  us  arrange 
the  names  U,  R,  Qy  My  T  in  the  following  order: 
M,  Qy  Ry  T,  U\  and  temporarily  attach  these  names 

as  follows :  M  to  X*  Q  ^^  <t>y  ^  ^^  ^*  ^  to  cr,  (/  to  k. 
Thus  the  set  of  names  have  to  be  attached  in  a  fixed 
order,  one  by  one,  to  the  set  of  things  taken  in  any 
order.  What  is  logically  required  to  avoid  mistake  is 
that  the  enumeration  of  the  things  should  be  both  ex- 
haustive and  non-repetitive — a  condition  which  children 
and  savages  often  find  difficult  to  fulfil.  Now,  inasmuch 
as  the  number-names  My  Q,  Ry  T,  6^  are  always  attached 
in  an  invariable  order,  the  last  number  named  indicates 
unequivocally  the  number  of  the  counted  set  of  objects. 
In  other  words,  the  cardinal  number  of  any  enumerable 
set  of  objects  is  unambiguously  indicated  by  the  ordinal 
number  of  the  correlated  number-names.  Histoncaiiy 
the  letters  of  the  alphabet,  having  been  memorised  in 
a  fixed  order,  served  also  as  the  written  symbols  for 
numbers;  but  their  employment  for  this  purpose  coukl 
not  be  extended   to  all  numbers,   since  an  alphabet 


If 


158 


CHAPTER  VII 


necessarily  consists  of  a  limited  number  of  letters. 
Moreover  it  is  psychologically  impossible  to  memorise 
an  endless  list  of  names.  Hence  it  was  necessary  to 
invent  some  system  which  would  render  it  possible  to 
count  any  set  of  things,  however  large.  The  Roman, 
Greek  and  Hebrew  alphabets  were  employed  for  this 
purpose  with  more  or  less  success,  but  were  finally 
superseded  by  the  Arabic  notation  in  which  place-value 
was  given  to  the  symbols  i,  2,  3,  4,  5,  6,  7,  8,  9,  and 
the  symbol  o  was  added.  These  ten  symbols  serve  as 
proper  names  of  numbers,  all  other  numbers  being  ex- 
pressed by  names  constructed  out  of  these.  Thus  the 
compound  word  twenty-four  or  the  compound  symbol 
24  is  analysable  as  meaning  'two  tens  plus  four,*  and 
therefore  to  be  understood  in  terms  of  the  operations 
of  multiplication  and  addition.  Such  compound  symbols 
or  words  are  not  proper  names  of  numbers  like  two,  ten, 
or  four,  but  may  be  called  constructed  names.  The 
elementary  learner  of  arithmetic  must,  in  fact,  reverse 
the  logical  order  of  thought,  and  understand  the  pro- 
cesses of  multiplication  and  addition  before  he  can 
intelligently  learn  to  count  beyond  twenty  or  so,  or 
understand  what  is  known  as  the  decimal  system  of 
notation. 

§  4.  We  now  pass  from  the  psychological  analysis  of 
counting  to  the  consideration  of  its  underlying  logical 
principles.  Counting  is  a  special  case  of  one-one  corre- 
lation, the  peculiar  characteristics  of  which  are  (i)  that 
a  prescribed  set  of  name-items  have  to  be  memorised 
in  a  definite  serial  order;  and  (2)  that  the  correlations 
are  factitious.  As  regards  (i)  the  mental  process  of 
counting,  which  involves  order,  must  be  contrasted  with 


i 


\ 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        159 

one-one  correlations  in  general  which  are  irrespective 
of  order.  As  regards  (2)  factitious  correlations  must  be 
contrasted  with  such  correlations  as  husband  and  wife, 
denominating  and  denominated  by  (in  an  ideal  language) 
etc.,  in  which  any  given  husband  is  correlated  with  a 
determinate  wife,  or  any  given  proper  name  with  a 
determinate  thing.  Now  in  my  view,  factitious  corre- 
lations are  essentially  necessary  in  x}i\^  general  ^^ci^orv  of 
numerical  equality;  though  they  never  enter  into  the 
abstract  deductions  of  arithmetic.  On  the  necessitv  of 
factitious  correlations,  recognised  authorities,  above  all 
Mr  Russell,  are  opposed  to  me.  Their  definition  of  the 
numerical  equality  of  two  sets  of  things  is,  in  effect, 
formulated  as  follows:  *  There  is  a  one-one  relation  of 
any  member  of  the  one  set  to  some  member  of  the 
other  set.'  But  it  seems  to  me  essential  to  distinguish 
the  statement  that  'the  items  can  be  correlated  one 
to  one'  from  the  statement  'there  is  a  one-one  correla- 
tion'; the  former  points  to  a  factitious,  the  latter  to  a 
factual  correlation.  There  need  be  no  relation  at  all  de- 
pending on  the  nature  of  the  items  themselves  comprised 
in  the  two  sets,  that  would  determine  which  item  of  the 
one  set  should  be  attached  to  any  given  item  in  the 
other.  If  relations  are  treated  extensionaily,  i.e.  as  mere 
substantive-couples,  then  it  is  of  course  a  matter  of 
fact  that  two  numerically  equal  classes  contain  couples 
of  items,  one  of  which  is  comprised  in  the  one  class  and 
the  other  in  the  other;  but  I  know  of  no  sense  in  which 
the  two  members  of  the  couple  are  related  the  one  to 
the  other,  except  that  the  one  is  temporarily  attached 
in  thought  by  some  thinker  to  the  other.  Apart  from 
this  factitious  coupling,  there  is  no  one-one  relation. 


V 


11 

i 


i6o 


CHAPTER  VII 


subsisting  between  any  given  item  of  the  first  set  and  any- 
determinate  item  of  the  second,  that  would  not  equally 
subsist  between  the  given  item  and  any  other  item  arbi- 
trarily selected  from  the  second  set.  Thus  the  establish- 
ment of  numerical  equality  between  two  finite  classes 
requires  in  general  factitious  correlations.  On  the  other 
hand,  the  only  mode  of  establishing  numerical  equality 
between  infinite  classes  is  to  discover  factualf  or  more 
specifically  formal,  correlations.  The  formal  correla- 
tions required  in  pure  arithmetic,  finite  and  transfinite, 
are  what  may  be  called  functional ;  and,  for  the  purposes 
of  this  elementary  exposition  of  the  logic  of  arithmetic, 
the  notion  of  functional  correlation  must  be  introduced 
and  explained. 

§  5.  Using  the  symbol/*  for  any  function  and /" for 
its  converse,  the  relation  n  to/{n)  will  be  one-one;  pro- 
vided  that  n  determines  uniquely  the  value  of/{n)  and 
f(ni)  determines  uniquely  the  value  oi  f\f{m)\.  For 
example:  \eX.f{n)  stand  for  ^  +  7,  then y*(;^^)  will  stand 
iox  m  —  T  \  and  the  integers  from  i  to  ;^  (inclusive)  can 
be  correlated  one  to  one  with  the  integers  from  8  to 
n-\-']\  each  integer  in  the  second  series  being  given  by 
adding  7  to  the  corresponding  integer  in  the  first,  and 
each  in  the  first  series  by  subtracting  7  from  the  corre- 
sponding integer  in  the  second.  Similarly,  \if(n)  stands 
for  «  X  7,  theny*(w)  will  stand  for  m-^  7 ;  and  the  integers 
from  I  to  w  can  be  correlated  one  to  one  with  the  multi- 
ples of  7  from  7  to  7«.  In  general :  if  the  relation  of  n  to 
f(n)  is  one-one,  then  the  series  of  values  assumed  by  n  is 
numerically  equal  to  the  series  of  values  assumed  by/"(;^). 

An  important  application  of  this  theorem  is  to  the 
case  where  the  integer  n  assumes  all  possible  finite 


\\ 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        161 

values,  obtained  from  unity  by  the  successive  addition 
of  unity.    In  this  case,  the  simplest  illustration  is  afforded 
by  taking /(;^)  to  stand  for  2n,  There  is  then  established 
a  one-one  correlation  of  the  successive  integers  i,  2, 
3,  4  ...  with  the  successive  integers  2,  4,  6,  8  ....    In 
other  words,  the  number  of  finite  integers  is  the  same 
as  the  number  of  finite  even  integers;  although   the 
former  series  comprises  all  the  odd  integers  and  these 
are  not  comprised  in  the  latter.    Thus,   although  the 
^gg^^g^te  of  even  integers  is  a  part  proper  of  or  sub- 
included  in  the  aggregate  of  integers,   yet  the  two 
aggregates  are  numerically  equal.    Now  we  may  define 
an  infinite  number  as  the  number  of  any  aggregate  that 
includes  a  part  proper  numerically  equal  to  itself.    Thus 
the  instance  above  cited  is  the  simplest  of  the  many 
proofs  that  establish  the  theorem  that  the  number  of 
finite  integers  is  infinite.    If  the  integers  are  presented 
in  ascending  order  of  magnitude,  the  series  so  conceived 
has  a  first  but  no  last  term  and  also  is  discrete  in  the 
sense  that  each  term  has  one  and  only  one  immediate 
successor.    The  cardinal  number  of  any  aggregate  that 
can  be  so  arranged  in  a  series  is  called  No-   This  is  the 
smallest  of  infinite  cardinal  numbers. 

The  reader  must  here  be  referred  to  the  mathe- 
matical exponents  of  the  theory  of  transfinite  cardinals 
and  ordinals  for  further  instruction.  The  most  com- 
prehensive account  of  this  theory  will  be  found  in 
Mr  Bertrand  Russell's  work  entitled  Principles  of 
Mathematics, 

§  6.  As  number  and  the  magnitudes  that  are  derived 
solely  from  number  may  be  called  abstract,  so  those 
which  contain  a  material  factor  may  be  called  concrete 


j.L.n 


II 


i52  CHAPTER  VII 

magnitudes  or  quantities.    Thus  duration-magnitudes, 
stretch-magnitudes,  magnitudes  of  qualitative  difference 
are  quantities,  because  the  entities  of  which  they  are 
predicable  are  defined  and  differentiated  in  terms  that 
are  not  purely  logical.    This  use  of  the  term  quantity 
differs  from  that  expressly  enjoined  by  Mr  Russell,  who 
defines  a  quantity  as  *an  instance  or  specification  of 
magnitude.'    He  then  proceeds  to  identify  the  relation 
thus  indicated  in  some  cases  with  that  of  substantive  to 
adjective,  and  in  others  with  that  of  determinate  to  de- 
terminable; whereas,  in  the  common  language  of  mathe- 
matics, quantity  stands  to  magnitude  in  the  relation  of 
species  to  genus,  with  which  my  use  of  the  term  quan- 
tity corresponds.    With  regard  to  quantities  the  three 
differentiae  which  I  hold  to  be  fundamental  or  primitive 
are  extensive,  distensive  and  intensive.    The  term  dis- 
tensive  magnitude  is  new,  and  the  reason  for  placing  it 
intermediarily  between  extensive  and  intensive  is  that 
by  some  logicians  it  has  been  included  under  extensive 
and  by  others  under  intensive  magnitude. 

An  extensive  magnitude  may  be  defined  as  one 
which  can  be  predicated  only  of  an  entity  that  can 
appropriately  be  called  a  whole.    The  notion  of  whole 
is  correlative  to  the  notion  of  part;  and,  more  precisely, 
a  whole  is  to  be  conceived  as  having  parts  which  can 
be   specifically   identified   and  distinguished  indepen- 
dently  of  their  relations  of  equality  or  inequality ;  e.g. 
a  finite  line  is  a  whole  of  the  simplest  possible  kind, 
under  the  figure  of  which  all  one-dimensional  wholes 
may  be  metaphorically  pictured.    Thus  a  line  CEG  is 
represented  as  having  the  parts  CE  and  EG,  each  of 
which  is  definitely  identifiable  for  itself  and  distinguish- 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        163 

able  from  the  other.    The  several  parts  of  a  whole  are 
of  the  same  nature  as  the  whole,  and  therefore  the  con- 
struction of  a  whole  out  of  parts  or  the  division  of  a 
whole  into  parts  may  always  be  called  homogeneous \ 
The  term  extensive  magnitude  has,  in  fact,  been  popu- 
larly restricted  to  spatial  and  temporal  wholes;  but  I 
shall  follow  Mr  Russell  in  applying  this  term  also  to 
certain  qualitative  wholes,  e.g.  to  a  continuous  aggre- 
gate of  hues  or  of  pitches.   Thus  we  speak  of  a  scale  of 
hue  and  a  scale  of  pitch  in  the  sense  of  a  class  com- 
prising all  specific  hues  or  pitches  which  are  qualita- 
tively intermediate  between  two  terminal  huesor  pitches. 
Now  the  class  comprising  such  determinate  items  con- 
stitutes what  is  now  called  a  stretch  ;  thus  a  qualitative  ^ 
stretch  of  hue  or  of  pitch  is  formally  analogous  to  the 
period  comprising  all  determinate  instants  between  one 
instant  and  another  or  to  the  geometrical  line  comprising 
all  points  intermediate  between  one  point  and  another. 
§  7.    It  might  appear,  since  the  instants  comprised 
in  a  period  and  the  points  comprised  in  a  \\nt  are  sub 
stantival,  while  the  hues  or  pitches  comprised  in   a 
qualitative  stretch  are  adjectival,  that  there  is  some 
fundamental  logical  distinction  between  these  two  kinds 
of  stretches.    Thus  :  though  either  may  be  metaphori- 
cally represented  by  a  line  CEG,  yet,  if  the  points  C, 
E,  G  stand  actually  for  points  or  instants — these  being 
substantival — the  stretch  represented  is  substantival ; 
whereas  if  C,  E,  G  represent  three  pitches — pitches 

^  The  term  whole  is  frequently  applied  to  a  construct  constituted 
of  heterogeneous  elements,  e.g.  to  a  proposition ;  but  for  such  a  con- 
struct the  term  unity  is  preferable,  unity  being  the  genus  of  which 
whole  is  a  species. 

II— 2 


164 


CHAPTER  VII 


being  characteristics  of  sound  and  therefore  adjec- 
tival— the  stretch  itself  is  adjectival.  It  is  open  to 
question,  however,  whether  points  or  instants  are  of  a 
substantival  nature ;  and  this  has  been  a  matter  of 
frequent  philosophical  dispute.  If  we  regard  time  and 
space  as  existents,  then  the  events  which  occur  at  a 
given  date  or  occupy  a  given  period,  like  the  ink-spots 
which  may  be  placed  at  different  points  or  the  ink- 
lines  which  may  be  drawn  on  paper,  have  as  substan- 
tives a  unique  kind  of  relation  to  the  substantives  of 
a  different  category — date,  period,  point  or  line — to 
which  they  are  attached.  Such  a  relation,  like  that  of 
characterisation,  is  unique  ; — in  the  sense  that  one  of 
its  terms  necessarily  belongs  to  a  certain  category  and 
the  other  to  a  certain  other  category.  The  relations 
*  occupying '  and  '  occurring  at '  further  resemble  the 
characterising  tie  in  being  unmodifiable  ;  thus,  of  any 
given  date  and  any  given  event  the  only  relevant  as- 
sertion that  can  be  made  is  that  the  event  either  did 
or  did  not  occur  at  that  date.  In  saying  of  an  event 
that  it  occurs  at  a  certain  date  or  of  a  material  body 
that  it  occupies  a  certain  region,  the  predications  may 
be,  not  ultimately  analysable  into  definable  relations 
to  a  definable  period  or  region,  but  regarded  rather 
as  adjectivally  unanalysable.  '  Occurring  at '  and  'occu- 
pying' are  therefore  properly  speaking  ties.  It  is  not, 
however,  formally  incorrect  to  regard  them  as  relations, 
in  the  same  way  as  we  have  allowed  characterisation  to 
be  analysed  as  a  relation  involving  the  two  correlatives 
characterising  and  characterised  by.  From  this  discus- 
sion it  will  be  seen  that  I  incline  to  the  view  that 
instants  of  time  and  points  of  space,  as  well  as  time 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        165 

and  space  as  wholes,  are  not  substantival  or  existential 
but  merely  adjectival. 

Now,  under  the  head  of  the  relativity  of  time  and 
space  two  distinct   philosophical    problems  are  often 
confused.    The  view  that  position  in  space  or  time  is 
definable  not  as  absolute,  but  only  as  relative  to  other 
points  or  instants  is  to  be  distinguished  from  another 
view  according  to  which  temporal  and  spatial  relations 
are  relations,  not  between  entities  such  as  points  and 
instants,  but  between  what  occupies  the  points  or  in- 
stants.  The  first  of  these  two  problems  is  appropriately 
described  as  the  question  of  the  absoluteness  or  rela- 
tivity of  time  and  space  ;  the  second  as  the  question  of 
the  substantival  or  adjectival  nature  of  time  and  space. 
In  the  Principles  of  Mathematics  Mr  Russell  explicitly 
maintains  the  absolute  view  as   regards   both    these 
problems ;  he  deliberately  asserts  that  position— a  term 
conveniently  used  both  for  space  and  time — is  absolute 
and  not  merely  relationally  definable  in  terms  of  other 
points  or  instants  ;  and  also  that  points  and  instants 
are  existents.    Now,  in  the  foregoing  analysis,  I   have 
taken  the  relative  i.e.  adjectival  view  on  the  second  of 
these  two  problems,  while  not  rejecting  the  absolute 
view  on  the  first.    The  adjectival  view  of  space  and 
time,   in  which  we   deny   such    separable   entities  as 
instants  and  points,  must  not  be  confounded  with  the 
class-view  :  that  identity  of  dating  merely  means  being 
comprised  in  a  certain  assigned  class  of  contempora- 
neous events.    For,  in  holding  that '  occupying  a  certain 
instant'   is  an   unanalysable   adjectival    predicate,   we 
maintain  at  the  same  time  that,  qua  predicate,  it  is  an 
identifiable  entity,  in  the  same  way  as  the  adjective 


i66 


CHAPTER  VII 


'  red '  is  an  identifiable  entity  when  predicated  now  of 
this  patch  and  then  again  of  some  existentially  other 
patch.  This  view  is  not  inconsistent  with  my  previous 
analysis  ;  for  I  have  repeatedly  maintained,  particularly 
in  my  analysis  of  the  principle  of  abstraction,  that  ad- 
jectival identity  cannot  be  resolved  merely  into  mem- 
bership of  a  certain  definable  class.  My  contention  for 
the  adjectival  nature  of  space  and  time  amounts  to  the 
statement  that  instants  and  points  are  substantival 
myths.  It  is  not  necessary,  however,  for  the  purposes 
of  this  exposition,  to  press  the  question  of  the  substan- 
tiality of  time  and  space,  for  any  difference  of  view  on 
this  point  does  not  affect  the  further  development  of 
the  subject. 

§  8.  Having  shown  the  analogies  between  the  three 
kinds  of  stretches — qualitative,  temporal,  and  spatial 
or  rather  linear — we  will  now  compare  such  extensive 
wholes  with  classes  considered  in  extension,  which  may 
be  called  extensional  wholes.  It  is  not  a  mere  accident 
of  language  that  the  term  extension  has  two  applica- 
tions in  philosophy,  these  generally  occurring  in  such 
different  contexts  that  they  are  not  confused.  But  it  is 
worth  while  drawing  attention  to  the  double  use  of  the 
word  ;  and,  in  so  doing,  to  examine  a  topic,  prominent 
in  modern  mathematics,  concerning  the  formal  agree- 
ments and  differences  between  extensional  and  exten- 
sive wholes.  An  extensional  whole,  otherwise  a  class, 
is  naturally  associated  with  the  notion  of  assignable 
items  of  which  the  class  is  composed ;  on  the  other  hand, 
a  linear  whole — which  illustrates  an  extensive  whole  of 
the  simplest  kind — is  apprehended  as  a  whole  without 
thinking  of  the  points  it  contains.     In  other  words  : 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        167 

the  items  in  an  extensional  whole  are  prior  in  thought 
to  the  whole,  which  appears  to  be  the  product  of  a 
constructive  process ;  while  the  extensive  whole  is 
prior  to  any  conception  of  points,  which  seem  to  be  the 
result  of  a  similar,  but  reversed,  process  of  thought-con- 
struction. This  psychological  distinction  Mr  Russell 
seems  to  regard  as  philosophically  negligible  ;  and  he 
devotes  a  large  part  of  his  exposition  to  a  proof  of  the 
essential  sameness  in  nature  of  extensional  and  ex- 
tensive wholes.  This  question  raises  the  same  problem 
as  that  discussed  by  Hume  and  Kant — the  former  in 
his  quarrel  with  the  mathematicians,  and  the  latter  in 
his  solution  of  the  antinomies. 

Let  us  then  examine  what  common-sense  would  elicit 
from  a  consideration  of  these  two  kinds  of  wholes.  With 
regard  to  extensional  wholes,  I  have  adopted  the  term 
*  comprise '  to  represent  the  relation  of  a  class  to  any 
of  its  items  or  members,  and  *  include*  to  represent  the 
relation  of  a  genus  to  any  of  its  species ;  and  it  is  of 
the  first  importance  to  note  that,  for  extensive  wholes, 
an  analogous  distinction  holds  between  the  relation  of 
a  line  to  any  of  its  points  and  the  relation  of  a  line  to 
any  of  its  parts  which  are  themselves  linear.  For  just 
as  a  class  comprises  items  which  have  to  one  another 
the  sole  relationship  of  otherness,  so  a  line  comprises 
points  which  have  to  one  another  the  sole  relationship 
of  otherness ;  and  again,  just  as  members  of  a  species 
are  members  of  the  genus,  so  points  in  a  linear  part 
are  points  in  the  linear  whole.  Further,  since  a  line  or 
stretch  contains  parts  in  the  same  sense  as  a  genus 
includes  species,  it  follows  that  such  purely  logical  or 
formal  relations  as  overlapping,  includent,  excludent, 


(  \ 


i68 


CHAPTER  VII 


which  apply  to  classes  and  not  to  items,  apply  also  to 
the  parts  of  a  stretch  but  not  to  points.  Now  the  parts 
into  which  a  three-dimensional  space  can  be  divided 
are  three-dimensional,  and  have,  qua  three-dimensional, 
all  the  properties  of  the  whole ;  similarly  the  parts  of 
a  two-dimensional  space  are  two-dimensional ;  and  the 
parts  of  a  one-dimensional  space  one-dimensional.  On 
the  other  hand,  of  the  contiguous  parts  of  a  three- 
dimensional  whole  the  common  boundary  is  two-dimen- 
sional ;  of  the  contiguous  parts  of  a  two-dimensional 
whole  the  common  boundary  is  one-dimensional ;  and 
of  the  contiguous  parts  of  a  one-dimensional  whole 
the  common  boundary  is  zero-dimensional,  i.e.  a  point. 
Restricting  our  discussion  to  the  last  case,  we  note  a 
very  substantial  difference  between  extensive  wholes 
and  extensional  wholes ;  for  within  a  merely  extensional 
whole  there  are  no  relations  of  contiguity,  whereas 
every  extensive  whole  is  apprehended  as  containing 
parts  which  are  either  literally  or  metaphorically  further 
from  or  nearer  to  one  another.  Hence  the  notion  of  a 
point  as  a  boundary  comprised  in  neither  or  in  both  of 
the  two  parts  of  a  line  has  no  analogy  amongst  members 
of  a  genus  which  belong  either  to  one  species  or  to 
another  and  cannot  belong  to  both.  It  further  follows 
that  an  extensive  whole  resembles  a  serial  or  ordered 
set  of  items  rather  than  a  mere  unordered  class  or 
enumeration. 

§  9.  Having  considered  the  nature  of  an  extensive 
whole,  i.e.  that  of  which  extensive  magnitude  may  be 
predicated,  we  will  pass  to  the  consideration  of  the 
kinds  of  entities  of  which  distensive  or  intensive  mag- 
nitude can  be  predicated.    By  distensive  magnitude  is 


<» 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        169 

meant  degree  of  difference,  more  particularly  between 
distinguishable  qualities  ranged  under  the  same  deter- 
minable \  Thus  the  difference  between  red  and  yellow 
may  be  greater  or  less  than  that  between  green  and 
blue ;  and  similarly  the  difference  between  the  pitches 
C  and  F  may  be  greater  or  less  than  that  between  B 
and  G,  The  notion  of  difference  is  apt  to  be  associated 
with  the  arithmetical  process  of  addition,  for  which  the 
term  *  addendum'  or  'subtrahend'  may  be  used  in  order 
to  distinguish  it  from  a  distensive  magnitude.  Thus  it 
is  preferable  to  say  that  successive  terms  forming  an 
arithmetical  progression  are  obtained  by  a  constant 
addendum,  just  as  those  forming  a  geometrical  pro- 
gression are  obtained  by  a  constant  multiplier.  This 
reference  to  arithmetical  and  geometrical  progressions 
is  needed  because  the  measure  of  qualitative  difference, 
in  its  logical  and  even  its  philosophical  sense,  is  in 
some  cases  or  on  some  grounds  to  be  conceived  as  an 
addendum,  and  in  other  cases  or  on  other  grounds  as  a 
multiplier.  For  example,  if  a  series  of  colours  are  pre- 
sented as  in  the  spectrum,  in  a  continuous  spatial  order, 
we  might  conceive  the  magnitude  of  difference  between 
any  one  hue  and  any  other  to  be  proportional  to  the 
length  in  the  spectrum  between  the  two  hues.  In  this 
case,  by  taking  any  hue  as  origin,  say  O,  such  that  A 
is  between  O  and  B^  and  representing  the  difference 
between  A  and  B  by  the  symbol  AB,  we  should  assume 
that  its  value  was  given  by  the  equation^  iff  =  OB  —  OA, 
On  the  other  hand,  as  regards  the  scale  of  pitch,  the 
scientist  would  naturally  connect  the  pitches  with  the 
physical  process  of  aerial  vibration,  and  measure  each 

^  See  Part  I,  p.  191. 


170 


CHAPTER  VII 


pitch  by  the  number  of  vibrations  per  second.  On  this 
assumption  the  *  difference '  between  C  and  G  would 
be  represented  by  the  ratio  of  f,  and  that  between  G 
and  ^  by  ^,  and  therefore  the  *  difference'  between  C 
and  B  would  be  f  x  |^  =  -y-.  These  two  examples  of  the 
two  natural  modes  of  estimating  degrees  of  qualitative 
difference — viz.  by  an  addendum  or  by  a  multiplier — 
are  typical  of  all  problems  regarding  distensive  or  even 
intensive  magnitudes. 

It  will  be  important,  however,  to  contrast  either  of 
these  more  physical  modes  of  conceiving  distensive 
magnitude  with  the  mode  that  has  become  familiar  to 
psychologists  ever  since  Fechner's  and  Weber's  experi- 
ments. According  to  Fechner  it  would  appear  that  the 
magnitude  of  difference  between  the  qualities  or  inten- 
sities of  sensations  should  be  determined  by  taking  as 
unit-difference  that  which  is  just  discernible  in  an  act 
of  perception  directed  to  the  sensations  as  experienced. 
It  should  be  here  noted  that  we  are  measuring  psychical 
entities,  and  not,  as  in  the  previous  discussion,  their 
physical  correlates.  Fechner  adopted  the  view  that  the 
proper  sensational  magnitude,  either  of  qualitative  or 
of  intensive  difference,  was  obtained  by  addition,  in 
which  equal  units  were  those  which  were  just  per- 
ceptible. When  he  compared  the  resulting  sensational 
magnitude  with  the  magnitude  of  the  stimulus  as 
measured  physically,  he  concluded  that,  while  the  sensa- 
tions could  be  ranged  in  arithmetical  progression,  the 
corresponding  stimuli  would  form  a  geometrical  pro- 
gression. By  means  of  an  elementary  mathematical 
process  it  will  be  seen  that  this  formula  can  be  ex- 
pressed by  saying  that  the  magnitude  of  the  sensation 


^» 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        171 

varies  as  the  logarithm  of  the  stimulus.  But  this  tech- 
nical development  is  not  our  concern  here.  I  wish 
rather  to  draw  attention  to  the  extraordinary,  and  in 
my  view  baseless,  assumption  that  the  just  discernible 
differences  at  the  different  points  in  a  scale  should  be 
taken  to  indicate  equal  adde^ida.  If  he  had  assumed 
what  appears  to  be  more  plausible  that  the  just  dis- 
criminable  qualities  were  those  which  bore  a  common 
ratio  to  one  another,  the  experimental  results  of  Fechner 
or  Weber  would  have  been  most  naturally  expressed 
in  the  formula  that  the  sensational  magnitude  (cr)  varies 
in  proportion  to  the  magnitude  of  the  physical  stimulus 
[s)  measured  from  a  certain  constant  (^0)  •  i-^« 

where  k  is  constant;  instead  of  by  the  formula 

a- =  ^  log  (^  -  ^o)- 
So  far  from  taking  discriminability  as  equivalent  to  an 
addendum,  it  is  more  plausible  to  consider  it  as  equiva- 
lent to  a  ratio\  For  example,  taking  the  visual  magni- 
tudes of  four  objects  A,  B,  C,  and  D,  if  we  can  just 
discriminate  between  the  magnitudes  of  A  and  B  and 
also  between  those  of  C  and  /?,  then  it  is  reasonable 
to  infer  that  the  ratio  of  B  to  A  is  equal  to  the  ratio 
of  D  to  C,  rather  than  that  the  addendum  by  which 
B  exceeds  A  is  equal  to  the  addendum  by  which  D 

*  I  am  not  here  concerned  with  the  accuracy  of  the  experiments 
made  by  Fechner,  nor  with  his  right  to  make  the  very  wide  induction 
from  the  artificial  nature  and  limited  number  of  cases  that  he  and 
his  successors  have  examined.  I  am  referring  merely  to  a  logical  and 
not  to  a  psychological  question,  namely,  the  justification  for  regarding 
our  power  of  discrimination  as  equivalent  to  our  power  of  perceiving 
additions  of  magnitude  rather  than  ratios. 


172 


CHAPTER  VII 


exceeds  C.  This  principle  for  measuring  psychical 
magnitude  may  be  applied  not  only  to  cases  of  direct 
sense-perception,  but  also  to  those  in  which  we  are 
guided  by  general  psychological  considerations ;  for 
example,  the  difference  of  pleasure  that  we  conceive 
to  be  produced  by  two  different  increases  of  income 
such  as  that  from  ^loo  to  ;^200  and  from  ;^iooo  to 
;^iioo  would  not  naturally  be  taken  to  be  equal ;  the 
increase  from  ;^ioo  to  ;^i  lo  would  rather  be  considered 
the  equivalent  of  the  increase  from  ;^iooo  to  ^iioo. 

§  lo.  We  now  pass  explicitly  to  the  third  funda- 
mental kind  of  magnitude,  namely  intensive,  which  has 
received  considerable  philosophical  attention.  Kant 
regarded  intensity  as  so  to  speak  equivalent  to  existence 
or  reality,  so  that  that  which  has  greater  intensity  has 
for  him  greater  reality.  The  point  in  which  this  view 
agrees  with  the  modern  theory  is  that  intensity  has  a 
terminus  in  the  value  called  zero ;  and  it  is  in  this 
respect  that  the  distinction  between  distensive  and 
intensive  magnitudes  is  most  clearly  marked ;  the  mini- 
mum or  zero  of  distensive  magnitude  is  identity,  whereas 
the  minimum  or  zero  of  intensive  magnitude  is  non- 
existence.  Another  obvious  distinction  between  the  two 
kinds  of  magnitude  is  that  distensive  magnitude  is  a 
relation  between  determinates  under  some  one  given 
determinable,  whereas  intensive  magnitude  holds  within 
each  separate  determinate,  or  even  amongst  different 
qualities  under  the  same  determinable.  Thus,  with  re- 
gard to  the  comparative  brightness  of  different  hues, 
we  may  predicate  equal  to,  greater  than  or  less  than, 
and  so  also  with  regard  to  the  loudness  of  sounds  of 
different  pitch.    It  is  impossible,  however,  to  compare 


4> 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        173 

two  kinds  of  intensive  magnitude  such  as  the  bright- 
ness of  a  light  sensation  with  the  loudness  of  a  sound 
sensation  ;  all  we  can  say  is  that  a  colour  of  zero 
brightness  would  be  non-existent,  and  a  sound  of  zero 
loudness  would  be  non-existent.  The  subtle  point  then 
arises  whether  the  notion  of  zero-intensity  of  sound  is 
distinguishable  from  the  notion  of  zero-intensity  of 
light.  In  popular  language  we  might  ask:  Is  there 
anything  to  distinguish  absolute  silence  from  absolute 
darkness  .'*  I  think  that  apart  from  an  organ  of  sensa- 
tion having  potentialities  as  a  medium  for  receiving 
sensations  we  must  say  that  zero-intensities  are  indis- 
tinguishable ;  it  is  only  through  the  capacity  of  visual 
and  auditory  imagery,  and  indirectly  through  the  pos- 
session of  organs  for  conveying  these  two  corresponding 
kinds  of  sensation,  that  distinctions  between  zeros  can 
have  for  us  any  import. 

§  II.  In  conclusion  I  have  to  explain  why  disten- 
sive magnitudes  have  been  confused  on  the  one  hand 
with  extensive  and  on  the  other  hand  with  intensive 
magnitudes.  As  regards  the  former,  the  confusion  is 
due  to  identifying  the  distensive  magnitude  of  differ- 
ence, say  between  the  pitches  C  and  Gy  with  the  stretch 
including  all  the  intermediary  pitches.  This  stretch 
illustrates  what  we  have  called  an  extensive  whole ; 
and,  in  so  far  as  it  can  be  measured,  its  measure  would 
be  equivalent  to  that  of  the  difference  between  C  and 
G\  i.e.  its  measure  would  be  equivalent  to  that  of  a 
distensive  magnitude,  but  the  natures  of  the  two  are 
non-equivalent.  As  regards  distensive  and  intensive 
magnitudes,  these  agree  in  so  far  as  they  both  apply  to 
qualities,   and   not  obviously  to   things    occupying   a 


174 


CHAPTER  VII 


quantum  of  space  or  time  or  forming  a  linear  or  tem- 
poral series ;  but  it  is  necessary  to  distinguish  them 
inasmuch  as  distensive  magnitude  requires  the  funda- 
mental conception  of  different  qualities  which  are  yet 
comparable;  while  intensive  magnitude  requires — what 
has  sometimes  been  paradoxically  described  as  the 
conception  of  a  thing  as  merely  qualitative,  and  yet  as 
susceptible  of  quantitative  variation. 

§  12.  Having  distinguished  different  kinds  of  mag- 
nitude, we  have  now  to  consider  how  magnitudes  of 
any  given  kind  are  to  be  compared ;  and  we  will  begin 
by  the  simplest  kind  of  magnitude,  viz.  that  which  can 
be  predicated  of  a  linear  whole. 

Mr  Russell  deliberately  adopts  the  view  that  the 
ultimate  parts  of  a  line  are  points,  of  which  the  number 
may  be  assumed  to  be  2  exp  X,  whatever  be  the  magni- 
tude of  the  line.  In  other  words,  any  comparison  of 
one  line  with  another  in  regard  to  magnitude  depends 
upon  something  other  than  the  number  of  points  which 
the  lines  contain.  Hence  the  magnitude  of  an  exten- 
sive whole,  as  illustrated  by  a  line,  cannot  be  estimated 
in  terms  of  pure  or  abstract  number.  In  this  respect 
it  is  of  a  totally  different  nature  from  a  class,  the  mag- 
nitude of  which  is  entirely  determined  by  the  number 
of  items  it  comprises,  or  by  the  number  of  exclusive 
sub-classes  into  which  it  may  be  divided.  It  follows 
then  that  magnitude,  when  applied  to  an  extensive 
whole,  has  a  different  meaning  from  magnitude  when 
applied  to  an  extensional  whole.  For  what  I  have 
called  an  extensive  whole  Mr  Russell  uses  the  term 
*  divisible  whole,'  because  the  notion  of  dividing  is 
essential  to  our  conception  of  the  relation  of  part  to 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        175 

whole,  particularly  in  temporal  and  spatial  applications. 
But,  in  discussing  the  principle  required  for  comparing 
the  magnitude  of  one  line  with  that  of  another,  he  uses 
the  phrase  *  magnitude  of  divisibility.'  This  phrase  ap- 
pears to  me  unfortunate  inasmuch  as  it  conveys  no 
meaning :  entities  may  be  distinguished  according  as  they 
do  or  do  not  possess  the  quality  of  divisibility ;  and  the 
term  magnitude  is  of  course  required  when  we  discuss 
whether  one  thing  is  greater  or  less  than  another.  But 
I  fail  to  see  how  we  can  regard  one  line  as  being  greater 
than  another  on  the  ground  that  it  possesses  the  quality 
of  divisibility  in  a  higher  degree.  It  is  quite  certain 
that  the  number  of  parts  into  which  a  shorter  line  can 
be  divided  is  exactly  the  same  as  the  number  of  parts 
into  which  a  longer  line  can  be  divided ;  as  also  are 
the  number  of  points  in  the  one  and  in  the  other.  The 
term  'magnitude  of  divisibility*  therefore  appears  to 
me  merely  to  conceal  what  really  is  the  problem  in- 
volved in  comparing  things  having  extensive  magnitude ; 
namely  the  conception  of  equality  of  magnitude. 

§  13.  What  do  we  mean  by  the  question,  or  how 
can  we  test,  whether  one  given  line  or  surface  or  bounded 
three-dimensional  figure  is  greater  or  less  than  another  } 
Or  again  whether  one  stretch  of  hue  or  of  pitch  is 
equal  to  or  greater  than  another  ?  In  general,  for  two 
extensive  wholes  M  and  N  of  the  same  kind,  if  M  in- 
cludes but  is  not  included  in  N  it  will  be  agreed  that 
the  magnitude  of  M  is  greater  than  that  of  A^;  or 
briefly,  the  relation  of  superincident  to  subincident, 
whole  to  part  proper,  entails  the  relation  of  greater  to 
less.  But,  if  the  wholes  M  and  N  are  coexc/usive,  then 
no  such  test  of  equality  or  inequality  can  be  directly 


176 


CHAPTER  VII 


applied  ;  and  in  order  to  compare  their  magnitudes  in 
this  case,  we  must  be  able  to  find  parts  of  M  that  can 
be  equated  to  one  another  as  also  to  parts  of  N.  This 
would  provide  us  with  a  unit  magnitude,  in  reference 
to  which  the  magnitudes  of  M  and  N  could  be  numeri- 
cally compared.  If  we  further  assume  that  the  wholes 
satisfy  the  strict  criterion  of  continuity  as  defined  by 
Cantor,  then  the  series  of  numbers  rational  and  irra- 
tional will  provide  means  for  comparative  measurement 
of  all  such  magnitudes.  On  this  assumption  the  only 
problem  that  remains  is  the  provision  of  a  test  or 
definition  of  equality  amongst  unit  parts.  The  possi- 
bility of  such  a  test  must  be  separately  examined  for 
the  three  cases  of  spatial,  temporal  and  qualitative 
stretches.  As  regards  spatial  wholes  of  one,  two  or 
three  dimensions,  the  classical  method  is  that  of  super- 
position, the  validity  of  which  must  be  carefully  con- 
sidered. It  is  obviously  absurd  to  think  of  the  parts  of 
space  themselves  as  moving ;  and  hence  the  so-called 
method  of  superposition  can  only  have  practical  signi- 
ficance when  we  distinguish  the  material  occupant  of 
a  place  from  the  place  which  it  occupies.  When  the 
material  occupants  of  space  are  superposed  one  upon 
another,  and  the  boundary  of  one  is  coincident  with 
that  of  the  other,  they  are  said  to  be  conterminous ;  and 
when  the  boundary  of  one  is  subincident  to  that  of  the 
other,  they  may  be  said  to  be  partially  conterminous. 
Thus  the  outer  boundary  of  a  liquid  and  the  inner 
boundary  of  a  closed  receptacle  which  it  fills  are  coinci- 
dent ;  and,  in  this  case,  the  volume  occupied  by  the 
liquid  is  equal  to  the  volume  unoccupied  by  the  receptacle. 
Again,  if  any  two  bodies  have  a  common  two-dimen- 


'^ 


\    '  if 


t 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        177 

sional  boundary  (which  does  not  enclose  a  volume),  then 
the  boundary  of  the  one  has  the  same  area/  magnitude 
as  that  of  the  other ;  and,  if  two  bodies  have  a  common 
one-dimensional  boundary,  then  the  boundary  of  the  one 
has  the  same  linear  magnitude  as  that  of  the  other.    If, 
moreover,  several  different  bodies  can  in  either  of  these 
three  ways  be  made  conterminous  with  some  one  given 
body,  the  volume,  area  or  length  of  the  corresponding 
boundary  of  the  one  is  equal  to  that  of  the  other.  But  this 
predication  of  equality  assumes  that  the  volume  of  ihe 
receptacle,  or  of  the  areal  or  linear  boundaries  of  the 
superposed  bodies  remains  unchanged ;  and  the  assump- 
tion that  in  the  course  of  time  a  material  body  does  not 
change  its  spatial  magnitude  is  in  general  invalid  ;  hence 
there  is  no  literally  logical  justification   for  asserting 
equality  or  inequality  in  general,  either  with  respect  to 
the  same  body  in  different  places,  or  with  respect  to  the 
different  places  which  the    same    body   may  occupy. 
Science  in  this  case  relies  upon  the  constancy  (under 
unchanged  conditions)  of  the  volume  of  certain  bodies, 
and  uses  these  as  standards  by  which  the  changes  of 
volume  of  other  bodies  are  tested,     in  this  process,  we 
are  continually  acquiring  more  precise  knowledge  of 
causal  conditions;  but  the  final  justification  for  com- 
parisons of  spatial   magnitude  is  to   be  found  in   the 
coherency  or  consistency  with  which  the  systematisation 
of  measurements  and  the  construction  of  physical  laws 
can  be  developed.    The  conclusion  follows  then  that  no 
directly  logical  test  can  be  found,  and  we  must  be  satis- 
fied with  the  indirect  principle  according  to  wliich  com- 
prehensive universals  are  asserted  on  the  mere  ground 
that  they  do  not  lead  to  appreciable  inconsistencies. 


J.  L.  n 


12 


\ 


178 


CHAPTER  VII 


The  problem  of  temporal  magnitude,  like  that  of 
spatial  magnitude,  is  met  first  by  the  axiom  that  events 
that  are  conterminous  at  both  ends  have  the  same 
temporal  magnitude,  and  secondly  by  the  postulate  that 
under  identical  causal  conditions  equal  changes  occupy 
equal  lengths  of  time.  We  then  employ  some  physical 
process,  such  as  the  movement  of  the  hands  of  a  watch, 
in  which  the  mechanical  conditions  can  be  estimated 
with  the  closest  approximation  to  exactitude,  and  adopt 
as  standard  time-units  the  times  occupied  by  the  changes 
thus  effected.  Conversely,  where  equal  changes  are 
effected  during  unequal  times,  we  infer  that  the  causal 
conditions  are  not  identical.  In  all  temporal  changes, 
the  means  by  which  we  can  measure  such  changes 
as  equal,  itself  depends  upon  the  assumption  that  we 
can  measure  certain  spatial,  distensive  or  intensive 
magnitudes. 

Turning  now  to  qualitative  magnitudes,  we  have  to 
consider  by  what  method  stretches  of  hue  and  pitch  can 
be  quantitatively  compared.  If  we  agree  that  the  stretch 
from  ^  to  ^  is  equal  to  that  from  C  to  G^  in  a  scale  of 
pitches,  this  cannot  be  tested  by  any  such  method  as 
that  of  superposition,  for  there  is  no  distinction  here 
corresponding  to  that  between  the  place  which  is  occu- 
pied on  the  one  hand,  and  that  which  is  movable  and 
can  occupy  indifferently  one  place  or  another  on  the 
other  hand.  If  a  qualitative  stretch  has  magnitude,  this 
involves  the  assumption  that  stretches  of  the  same  kind 
are  comparable  as  greater  or  less.  But  how  much 
greater,  or  by  what  ratio  the  two  are  to  be  compared 
must  be  determined,  if  at  all,  by  some  principle  totally 
different  from  superposition.    Mathematicians  who  have 


1 


THE  DIFFERENT  KINDS  OF  MAGNITUDE       179 

written  on  this  subject  appear  to  agree  in  the  view  that 
two  magnitudes  may  be  comparable  as  greater  or  less, 
and  yet  not  measurable  in  terms  of  number.  But,  if  two 
stretches  are  mutually  excludent,  I  can  see  no  sense  in 
which  they  can  be  compared  as  greater  or  less,  unless 
we  have  a  test  of  equality;  and,  when  such  test  is 
forthcoming,  a  numerical  measurement  seems  to  me 
immediately  to  follow.  Numerical  measurement  is 
not  a  merely  arbitrary  one-one  correlation  between 
numbers  and  magnitudes:  for  such  correlation  could 
only  mean  that  for  the  greater  magnitudes  we  apply 
higher  numbers,  and  the  precise  numbers  which  we 
correlate  would  be  absolutely  arbitrary.  Hence  it 
appears  to  me  that  if  a  specific  one-one  numerical 
correlation  has  an  objective  ground,  according  to 
which  it  is  to  be  preferred  to  any  other,  this  must 
be  because  we  have  adopted  some  principle  for  de- 
termining a  correct  quantitative  unit  For  example, 
if  we  prefer  the  absolute  measurement  of  tempe- 
rature to  the  thermometric  measurement  as  deter- 
mined say  by  the  changes  of  volume  of  mercury,  this 
is  because  we  believe  that  the  differences  of  tempe- 
rature indicated  by  the  former  scale  do  correspond 
to  really  equal  differences  of  magnitude,  whereas  the 
other  does  not.  Readers  of  Clerk  Maxwell's  Heat 
will  learn  that  the  absolute  measurement  of  tempera- 
ture depends  upon  measurements  of  heat  and  work, 
which  are  complex  quantities,  being  partly  extensive 
and  partly  intensive.  In  all  such  cases,  where  we  can- 
not directly  measure  a  cause  or  an  effect,  we  measure 
it  indirectly  in  terms  of  its  effect  or  cause  (as  the  case 
may  be). 

12 — 2 


i8o 


CHAPTER  VII 


§  14.  The  entities  of  which  either  extensive,  disten- 
sive  or  intensive  magnitude  can  be  predicated  alone 
may  be  termed  simple  or  simplex,  and  from  these  kinds 
of  entity  we  now  pass  to  those  which  may  be  called 
compound  or  complex,  on  the  ground  that  two  or  more 
magnitudes  are  combined  in  our  conception  of  the 
quantity  of  the  resultant  complex.  These  latter  may  be 
illustrated  by  light  sensations,  which  vary  intensively 
according  to  their  brightness,  distensively  as  regards 
their  hue,  and  in  yet  a  third  respect  according  to  the 
proportion  in  which  the  chromatic  and  achromatic  factors 
are  combined  to  produce  different  degrees  of  saturation. 
Similarly  sound  sensations  vary  intensively  according 
to  their  loudness,  distensively  as  regards  their  pitch, 
and  as  regards  timbre  or  klang-tint  in  accordance  with 
the  proportional  intensities  of  their  constituent  tones, 
under-tones  and  over-tones.  It  is  convenient  to  speak, 
then,  of  light  and  sound  sensations  as  three-dimensional, 
in  the  sense  that  there  are  three  distinct  determinates 
under  which  any  such  sensation  can  be  defined  and 
quantitatively  estimated.  But  the  simplest  case  of  a 
three-dimensional  quantity  is  space.  In  space  we  may 
take  three  arbitrary  directions ;  and,  according  to  the 
ordinary  view,  the  magnitudes  (i.e.  lengths)  along  these 
directions  have  the  unique  characteristic  of  being  com- 
parable. Any  point  in  a  space  of  three  dimensions  is 
therefore  assignable  by  three  ordinates  drawn  in  deter- 
mined directions  from  a  given  point  as  origin.  In  this 
way  a  surface  in  three  dimensions,  or  a  line  in  two 
dimensions,  differs  from  what  is  called  a  graph,  in  that 
the  magnitudes  represented  by  the  ordinates  of  a  point 
in  the  graph  are  of  different  kinds  and  therefore  incom- 


t 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        181 

parable.  For  example:  a  graph  representing  the  co- 
variation of  work  done  and  hours  expended  uses  two 
incomparable  magnitudes. 

The  general  topic  which  we  have  now  to  consider 
is  that  of  a  derived  quantity  that  is  constructed  by  some 
kind  of  combination  of  other  quantities.  I  n  constructing 
a  quantity  comparable  with  each  of  those  combined,  the 
processes  of  addition  and  subtraction  can  alone  be 
applied;  and,  conversely,  addition  and  subtraction  can 
only  be  applied  to  comparable  quantities.  Such  addition 
and  subtraction  may  be  termed  concrete,  in  antithesis 
to  abstract  in  which  pure  numbers  are  concerned  whose 
sum  or  difference  is  also  a  pure  number.  Now  I  shall 
maintain  that  processes  analogous  to  multiplication  and 
division  may  be  employed  in  constructing  a  quantity  of 
a  different  kind  from  any  of  those  that  are  combined  in 
its  construction;  and  such  multiplication  or  division 
may  also  be  called  concrete.  Thus,  considering  first  the 
three  notions  of  length,  area,  and  volume,  I  shall  say 
that  the  multiplication  of  two  differently  directed  lengths 
constitutes  an  area,  and  that  of  three  differently  directed 
lengths  constitutes  a  volume.  Here  we  are  extending 
the  operation  called  multiplication  beyond  its  primary 
use.  For,  while  it  is  universally  agreed  that  we  may 
multiply  a  pure  number  by  a  pure  number,  in  con- 
structing another  pure  number,  or  a  quantity  of  any 
kind  by  a  pure  number,  in  constructing  a  quantity  of 
the  same  kind,  yet  most  mathematicians  have  refused 
to  allow  that  by  multiplying  one  quantity  by  another 
we  may  construct  a  third  quantity  different  in  kind  from 
both  the  quantities  multiplied.  They  maintain  that  what 
is  multiplied  is  the  numerical  measure  of  the  quantities 


.11 


182 


CHAPTER  VII 


and  not  the  quantities  themselves.  Similarly  with  regard 
to  division:  it  is  agreed  that  we  may  divide  a  pure 
number  by  a  pure  number  in  constructing  another  pure 
number,  or  a  quantity  of  any  kind  by  a  pure  number  in 
constructing  a  quantity  of  the  same  kind ;  but  we  are 
prohibited  from  dividing  one  quantity  by  another  in 
constructing  a  quantity  different  in  kind  from  the  quan- 
tities divided.  In  this  case  too,  the  so-called  division 
is  regarded  as  a  division  not  of  the  quantities  but  of 
their  numerical  measures.  My  first  objection  to  this 
view  is  that  it  offers  no  means  of  distinguishing  between 
the  multiplication  or  division  of  a  quantity  by  a  pure 
number,  which  yields  a  quantity  of  the  same  kind,  from 
that  very  different  kind  of  multiplication  or  division 
which  yields  a  quantity  different  in  kind  from  those 
multiplied  or  divided.  My  disagreement,  however,  with 
the  almost  unanimous  opinion  of  mathematicians  may 
perhaps  be  considered  merely  verbal ;  but  the  view  that 
I  maintain  is,  I  think,  based  upon  an  important  logical 
principle.  Apart  from  any  conception  of  numerical 
measurement  which  adopts  numbers,  integral,  rational 
and  irrational,  it  appears  to  me  that  we  must  conceive 
the  process  of  multiplying  say  a  foot  by  an  inch  (which 
involves  no  idea  of  number)  as  a  construction  by  which, 
from  two  magnitudes  of  the  same  kind,  a  third  magni- 
tude of  a  different  kind  is  derived.  If  no  such  magni- 
tude were  presented  to  perception  or  thought,  it  would 
follow  that  no  meaning  could  be  attached  to  such  multi- 
plication ;  but,  inasmuch  as  an  area  is  a  genuine  object 
of  thought  construction,  I  see  no  insurmountable  objec- 
tion to  speaking  of  the  process  of  multiplication  as  that 
by  which  area,  for  instance,  is  constructed  out  of  two 


# 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        183 

directed  lengths,  or  volume  out  of  three.    The  mathe- 
maticians who  reject  this  idea  hold  that  the  notion 
of  units  of  different  kinds  is  sufficient,  without  intro- 
ducing the  multiplication  or  division  of  units.    It  is 
agreed  that  the  area  of  a  rectangle  whose  sides  are  of 
unit  length  is  a  unit  area,  and  the  volume  of  a  cube 
whose  sides  are  of  unit  length  is  a  unit  volume.    In  this 
way  the  numerical  measures  of  area  and  volume  are  ob- 
tained by  multiplying  the  numerical  measures  of  their 
sides;  but  in  my  view  we  must  allow  that  the  lengths 
themselves  are  multiplied,  for  otherwise  we  could  not 
distinguish  the  different  kindsof  magnitudes  constructed, 
since  where  only  abstract  numbers  are  concerned  only 
abstract  numbers  are  constructed,  and  there  is  nothing 
to  indicate  the  difference  between  one  quantity  thus 
derived  and  another.    Passing  from  concrete  multipli- 
cation to  concrete  division  we  have  what  may  be  thought 
a  more  interesting  and  certainly  a  wider  application  of 
the  same  general  principle.    Those  quantities  which  are 
derived  by  dividing  one  kind  of  quantity  by  another 
may  be  called  rate-quantities,  or  in  certain  cases  degree- 
quantities.    A   rate-quantity   is   expressed   in   familiar 
English  by  the  Latin  word  per,  of  which  it  is  easy  to 
multiply  examples:   e.g.  space  traversed  per  second, 
wages  earned  per  hour,  pleasure  experienced  per  minute, 
pressure  per  square  foot,  mass  per  cubic  foot.    The  two 
constituent  quantities  in  this  kind  of  division  may  be 
themselves  complex  or  of  different  kinds,  extensive, 
distensive  or  intensive ;  but  so  far  as  the  conception  of 
concrete  division  is  concerned,  no  logical  distinctions 
are  required  in  analysing  the  general  notion  of  a  rate- 
quantity.    Each  of  the  rate-quantities  constructed  by 


i84 


CHAPTER  VII 


this  species  of  division  is  a  quantity  of  a  different  kind 
from  the  quantities  of  which  it  is  constituted;  and,  as  in 
multiplication,  it  is  always  useful  for  arithmetical  pur- 
poses to  adopt  as  the  derived  unit-quantity  that  which 
is  constructed  out  of  fundamental  unit-quantities.  The 
general  term  'rate'  which  I  have  introduced  is  in  common 
use :  thus  we  mean  by  the  rate  of  wages  the  quantum 
of  wages  earned  per  unit  of  time,  by  the  rate  of  speed 
the  quantum  of  length  traversed  per  unit  of  time,  and 
by  the  rate  at  which  pleasure  is  being  experienced,  the 
quantum  of  pleasure  per  unit  of  time — these  being  cases 
in  which  the  rate  is  estimated  in  reference  to  time. 
Again  the  rate  called  hydrostatic  pressure  is  the  quantum 
of  pressure  per  unit  of  area;  the  rate  called  density  is 
the  mass  per  unit  of  volume.  The  term  degree,  which 
is  sometimes  used  instead  of  rate,  is  ambiguous  inas- 
much as  it  is  often  used  as  equivalent  to  intensity ;  but 
the  terms  rate  and  intensity  or  degree  ought  to  be 
clearly  distinguished,  because  the  notion  of  intensity 
refers  to  a  single  determinate  quality,  whereas  rate  is 
always  constituted  out  of  two  distinguishable  quantities ; 
moreover  the  notion  of  rate,  which  involves  concrete 
division,  is  always  correlated  with  concrete  multiplica- 
tion. For  example  velocity,  i.e.  rate  of  movement,  which 
involves  the  division  of  space  by  time,  involves  the 
converse  process  of  multiplying  velocity  by  time  in 
constructing  space.  But  if  we  conceive  of  velocity 
only  by  its  numerical  measure,  confusion  results  be- 
tween an  abstract  number  on  the  one  hand  and  the 
very  many  different  kinds  of  quantity  on  the  other 
hand  that  may  be  measured  by  the  same  abstract 
number. 


\ 


u  ▼  * 


1. 


,  i.e.  [-^.[^"T;  momentum,  i.e.  mass x velocity 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        185 

§  15.  The  practical  importance  of  recognising  con- 
crete multiplication  and  division  is  best  indicated  by 
explaining  what  is  meant  by  the  algebraical  dimensions 
of  a  quantity.  We  have  already  spoken  of  dimensions 
in  its  geometrical  sense;  thus  an  area  is  of  dimension 
two  in  regard  to  length,  a  volume  of  dimension  three  in 
regard  to  length.  Symbolising  the  dimension  length  by 
\L\  that  of  area  is  symbolised  by  \_U\  and  that  of  volume 
by  [Z'].    Similarly  velocity,  i.e.  length  per  time,  is  di- 

mensionally  L-J  or  [Z]  .  [7"'];  acceleration,  i.e.  velocity 

per  time  is  [Z].  [7^"'];  density,  i.e.  mass  per  volume  is 

is  [^][Z]  [7"~^];  force,  i.e.  mass  x  acceleration  is 
[J/].[Z].[7"'];  hydrostatic  pressure,  i.e.  force  per 
area  is  [J/]  .  [Z"']  [T"'],  etc.,  etc.  Now  the  one  rule  as 
regards  dimensions  is  that  the  additions  and  subtractions 
that  are  involved  in  a  quantitative  equation  must  always 
operate  upon  homogeneous  quantities ;  i.e.  upon  quan- 
tities all  of  which  have  the  same  dimensions — these 
dimensions  being  generally  expressed  in  terms  of  the 
three  fundamental  incomparables  mass,  length,  and  time. 
Regarding  multiplication  and  division,  in  accordance 
with  my  view,  as  real  operations  performed  upon  con- 
crete quantities,  the  square  bracket  in  the  above  symbols 
stands  for  a  concrete  unit.    For  example  the  velocity 

320  ft.  __  i6ft.  _  £6     - 
60  sec.      3  sec.       3 
unit  velocity.     Those   mathematicians  who  hold  that 
such  an  expression  as  ft. -^  sec.  is  meaningless  have  to 
maintain  that  the  mathematical  equations  which  are 


*320  feet  per  60  seconds*  means 


^ 


i86 


CHAPTER  VII 


used  to  express  physical  facts  are  concerned  only  with 
the  numerical  measurement  of  concrete  quantities, 
whereas  I  hold  that  they  are  concerned  with  the  concrete 
quantities  themselves. 

§  1 6.    There  is  one  very  unique  case  in  concrete 
division,  viz.  where  the  dividend  and  divisor  are  quan- 
tities of  the  same  kind.    In  general  the  result  of  such 
division  is  to  construct  a  pure  ratio,  i.e.  a  magnitude 
which,  when  entering  as  multiplier  or  divisor  of  a  quan- 
tity of  any  kind  yields  a  quantity  of  the  same  kind,  like 
the  processes  of  addition  and  subtraction  of  quantities. 
But  when  a  length  is  divided  by  a  length,  or  an  area  by 
an  area,  we  often  intend  the  result  of  such  division  to 
represent  an  angle.    It  is  therefore  necessary  to  dis- 
tinguish those  cases  in  which  the  division  of  a  length 
by  a  length  represents  a  mere  ratio,  from  those  in 
which  it  represents  an  angle.    In  the  former  case,  the 
quotient  being  a  pure  number  can  be  used  as  a  multi- 
plier or  divisor  for  a  quantity  of  any  kind  whatever; 
but  in  the  latter  case  this  is  never  possible;  one  angle 
can  only  be  mathematically  combined   with   another 
angle,  and  this  only  by  the  operation  of  addition  or  of 
subtraction.    The  further  complication  in  respect  of  the 
measurement  of  an  angle  is  that  this  measurement  may 
be  used  in  different  algebraical  applications  alternatively 
either  as  an  abstract  ratio  or  as  a  concrete  quantity,  which 
is  denoted  by  the  term  angle.    But  the  special  question 
which,  in  my  view,  requires  a  clear  answer  is  how  to 
distinguish  the  process  of  dividing  length  by  length  that 
yields  a  mere  ratio,  from  what  appears  to  be  the  same 
process  and  yet  yields  an  angle.    The  answer  seems  to 
be  that  when  we  are  merely  comparing  two  lengths 


f 


I 


THE  DIFFERENT  KINDS  OF  MAGNITUDE        187 

which  may  be  said  to  be  dissociated,  their  comparison 
yields  a  mere  ratio,  while  when  connecting  two  asso- 
ciated lengths  in  the  process  of  division,  we  are  con- 
structing an  angle.  Thus,  when  we  define  the  magni- 
tude of  an  angle  by  the  ratio  of  the  arc  of  a  circle  to  its 
radius,  the  arc  and  the  radius  are  associated  in  our  con- 
ception of  the  mode  in  which  the  angle  is  constructed; 
but  when  we  are  merely  comparing  the  length  of  one 
line  with  thatof  any  other,  no  natural  association  between 
the  two  lines  is  involved.  The  same  holds  of  the  dif- 
ferential coefficient  dy  by  dx,  when  used  in  geometry 
to  represent  the  slope  of  a  tangent  of  a  curve,  which  is 
a  concrete  quantity  in  the  same  sense  as  the  quotient 
foot  by  second  representing  velocity. 

§  17.  To  sum  up:  Of  the  different  kinds  of  magni- 
tude, the  first  division  is  between  abstract  and  concrete, 
abstract  magnitudes  being  represented  by  pure  numbers, 
these  falling  into  the  three  divisions  of  integral,  rational 
and  irrational.  Amongst  concrete  quantities — namely 
those  that  involve  conceptions  obtained  from  special 
kinds  of  experience,  and  which  are  therefore  not  purely 
logical — we  distinguish  the  fundamental  or  primitive 
from  the  complex  or  derivative;  the  former  being  sub- 
divided into  extensive,  distensive  and  intensive  magni- 
tudes, out  of  which  the  various  derived  or  complex 
quantities  have  been  shown  to  be  constructed  by  opera- 
tions analogous  to  arithmetical  multiplication  and  divi- 
sion. These  complex  magnitudes  fall  again  into  different 
kinds,  the  distinctions  between  which  may  be  always 
indicated  by  expressing  the  quantity  dimensionally,  i.e. 
as  involving  a  concrete  product  of  different  fundamental 
quantities,  each  entering  with  a  positive  or  negative 


\ 


i88 


CHAPTER  VII 


index.  Finally  a  fundamental  distinction  has  been  drawn 
between  addition  or  subtraction  on  the  one  hand  and 
multiplication  or  division  on  the  other;  inasmuch  as  the 
quantities  added  or  subtracted  must  be  of  the  same  kind, 
i.e.  represented  as  dimensionally  equivalent;  whereas 
the  operations  of  multiplication  and  division  yield  a 
quantity  different  in  nature  from  its  factors,  which, 
however,  together  determine  its  nature.  Throughout 
the  whole  discussion  of  concrete  magnitudes,  the  diffi- 
cult problem  of  defining  or  testing  equality  has  been 
examined  for  each  fundamentally  distinct  kind  of  quan- 
tity. The  treatment  has  been  comparatively  elementary, 
the  reader  being  referred  for  more  subtle  distinctions 
and  analyses  to  works  which  deal  primarily  with  mathe- 
matics and  its  philosophy. 


V 

) 


I 


CHAPTER  VIII 

INTUITIVE  INDUCTION 

§  I.  Induction  in  general  may  be  contrasted  with 
deduction  in  that  for  a  universal  conclusion  deduction 
needs  universal  premisses,  whereas  in  induction  a  uni- 
versal conclusion  is  drawn  from  instances  of  which  it  is 
a  generalisation.  Here  the  emphasis  is  upon  the  word 
instances,  because  although  the  customary  account  of 
deduction  is  that  the  range  of  the  conclusion  is  identical 
with  that  of  the  narrowest  of  the  premisses,  yet  de- 
duction must  include  cases  in  which  the  range  of  the 
conclusion  is  not  identical  with  that  of  any  one  of  the 
premisses,  and  may  even  be  wider  than  the  widest  of 
them.  Actually  the  antithesis  between  inductive  and 
deductive  inference  is  not  so  fundamental  as  that  between 
demonstrative  and  problematic  inference ;  for  every 
form  of  induction,  except  the  problematic,  is  based  upon 
the  same  fundamental  principles  (and  these  alone),  as 
syllogism  and  other  forms  of  deduction ;  whereas  it  is 
impossible  to  establish  a  theory  of  problematic  induc- 
tion, without  recourse  to  certain  postulates  that  are  not 
involved  in  either  form  of  demonstration,  whether  de- 
ductive or  inductive.  Now  the  fundamental  principles 
which  underlie  demonstrative  forms  both  of  induction 
and  deduction  are  themselves  based  upon  a  kind 
of  inference  which  may  be  called  intuitive  induction. 
This  process  is  not  limited  to  the  establishment  of  the 
principles  of  demonstration,  but  applies  also  to  certain 
material  as  well  as  formal  generalisations. 


\ 


190 


CHAPTER  VIII 


INTUITIVE  INDUCTION 


191 


We  have  so  far  referred  to  two  types  of  induction, 
viz.,  intuitive  and  demonstrative ;  it  will  be  convenient 
to  distinguish  in  all  four  varieties,  namely  intuitive, 
summary,  demonstrative  and  problematic.  Of  these 
the  three  former  will  be  discussed  in  the  present  Part 
of  this  work,  but  problematic  induction  will  be  examined 
in  detail  in  a  separate  Part,  on  the  ground,  specified 
above,  of  its  dependence  upon  special  postulates. 

§  2.  Before  treating  the  main  topic  of  this  chapter, 
we  must  discuss  the  necessarily  preliminary  process 
known  as  abstraction,  the  nature  of  which  was  a  special 
subject  of  philosophical  and  psychological  controversy 
amongst  James  Mill  and  his  contemporaries.  The  dis- 
cussions of  that  date  started  from  the  supposition  that 
what  was  presented  in  our  earliest  acts  of  perception 
was  a  combination  of  impressions  from  different  senses, 
such  as  those  of  sight  and  touch.  From  this  pre- 
supposition, upon  which  both  parties  were  agreed,  the 
difficulty  was  raised  as  to  how  the  percipient  could 
single  out  an  occurrent  impression  of  one  sense  from 
the  concurrent  impressions  of  other  senses.  This  pre- 
supposition, however,  is  fundamentally  mistaken.  For, 
in  /act,  our  earliest  acts  of  attention,  which  yield  any  ^  \Ny 
product  that  could  be  called  a  percept,  are  directed  to  \  \  ^ 
impressions  of  one  sense  at  one  time,  and  to  impressions 
of  another  sense  at  another  time.  For  example,  the 
child  when  interested  in  the  colour  of  a  ball,  is  attending 
to  his  visual  impressions  apart  from  any  motor  or  tactual 
sensations  that  he  may  be  experiencing  in  handling 
the  ball ;  that  is  to  say,  his  attention  is  from  the  first 
exclusive,  and  it  is  only  in  further  progress  of  attentive 
power  that  his  attention  becomes  inclusive.    The  atten- 


r 


I 


tion  that  includes  visual  with  tactual  impressions  is  a 
higher  and  later  process  than  the  attention  which  is 
directed  either  exclusively  to  the  visual  impressions  or 
exclusively  to  the  tactual  impressions.  The  fact  that 
we  can  and  do  attend  to  impressions  of  one  order  in 
disregard  of  concurrent  impressions  of  other  orders, 
explains  how  our  primitive  perceptual  judgments,  from 
the  first,  assume  a  logically  universal  form.  For,  in 
predicating  a  determinate  colour,  for  instance,  of  any 
given  impression,  there  is  a  recognition  that  the  same 
determinate  can  be  predicated  of  a/l  impressions  which 
agree  with  the  given  impression  in  respect  of  colour, 
however  much  they  may  disagree  in  other  respects. 
Now,  if  this  be  granted,  it  has  an  important  bearing 
upon  another  serious  historical  controversy — namely 
that  between  Mill  and  his  opponents  as  to  the  founda- 
tions of  geometry.  Both  parties  to  this  dispute  started 
with  an  obscure  view,  that  there  was  an  opposition 
between  intuition  and  experience ;  whereas  in  truth 
intuition  is  a  /orm  of  knowledge,  in  relation  to  which 
experience  is  the  matter.  The  intuitionists  seem  to 
have  held  that  the  intuitive  form  of  knowledge  involved 
no  reference  to  experience ;  whereas  the  empiricists 
forgot,  when  relying  upon  experience  as  the  sole  factor 
in  knowledge,  that  knowing  is  a  mode  of  activity,  and 
therefore  not  of  the  same  nature  as  sense-experience 
which  is  merely  passive  or  recipient.  The  truth  is  that 
when  we  have  asserted  a  predicate  of  a  particular,  we 
have  apprehended  the  universal  in  the  particular,  in  the 
sense  that  the  adjective  is  universal  and  the  object  of 
which  it  is  predicated  is  particular. 

§  3.    There  is  another  sense  in  which  we  may  be 


\ 


192 


CHAPTER  VIII 


said  directly  to  apprehend  the  universal  in  the  par- 
ticular, namely  in  regard  to  certain  classes  of  proposi- 
tions, where  the  terms  universal  and  particular  apply 
to  the  propositions  themselves,  and  not  to  the  distinc- 
tion between  the  subject  and  the  predicate  within  the 
proposition.  It  is  at  this  stage  that  we  pass,  in  our 
discussion,  from  abstraction  to  our  main  topic,  viz., 
abstractive  or  intuitive  induction.  The  term  intuitive 
is  taken  to  imply  felt  certainty  on  the  part  of  the 
thinker  ;  and  it  is  characteristic  of  propositions  estab- 
lished by  means  of  intuitive  induction  that  an  accumu- 
lation of  instances  does  not  affect  the  rational  certainty 
of  such  intuitive  generalisations.  The  procedure  by 
which  these  generalisations  are  established  may  be 
shown  by  psychological  analysis  to  involve  an  inter- 
mediate step  by  which  we  pass  from  one  instance  to 
others  of  the  same  form  and  in  this  passage  realise 
that  what  is  true  of  the  one  instance  will  be  true  of  all 
instances  of  that  form. 

§  4.  Two  types  of  intuitive  induction  may  be  dis- 
tinguished, experiential  and  formal,  although  these  types 
are  not  precisely  exclusive  of  one  another. 

The  experiential  type  of  intuitive  induction  may  be 
illustrated  from  our  immediate  judgments  upon  sense- 
impressions  and  the  relations  amongst  them.  For 
example,  in  judging  upon  a  single  instance  of  the 
impressions  red,  orange  and  yellow,  that  the  qualitative 
difference  between  red  and  yellow  is  greater  than  that 
between  red  and  orange  (where  abstraction  from  shape 
and  size  is  already  presupposed)  this  single  instantial 
judgment  is  implicitly  universal ;  in  that  what  holds  of 
the  relation  amongst  red,  orange  and  yellow  for  this 


r 


) 


>) 


INTUITIVE  INDUCTION  193 

single  case,  is  seen  to  hold  for  all  possible  presenta- 
tions of  red,  orange  and  yellow.    Again  in  immediately 
judging  that  a  single  presented  object,  whose  shape  is 
perceived  to  be  equilateral  and  triangular,  is  also  equi- 
angular (where  abstraction  from  colour  and  size  is  pre- 
supposed) we  are  implicitly  judging  that  a//  equilateral 
triangles  are  equiangular.    Similarly  when  judging  for 
a  single  instance  that  the  sounds  A,  C,  F,  produced, 
say,  from  the  human  voice,  are  in  an  ascending  scale 
of  pitch,  we  are  implicitly  judging  that  all  sounds — 
apart  from  differences  of  timbre  or  loudness  such  as 
those  produced  by  the  violin  or  piano — that  can  be 
recognised  as  of  the  same  pitches  A,  C,  F,  are  also  in 
an  ascending  order  of  pitch.    The  universality  of  these 
experiential  judgments  extends  over  imagery  as  well 
as  sense  impressions  :  the  fact  that  we  can  identify  a 
specific  image  as  corresponding  to  a  specific  impression 
is  sufficient  to  enable  us  directly  to  transfer  our  judg- 
ments about  the  relations  amongst  impressions  to  those 
amongst  the  corresponding  images.    These  elementary 
illustrations  show  that  intuited  universals  about  colours 
and  pitches  are  of  the  same  epistemological  nature  as 
those  about  geometrical  figures,  in  that  the  judgment 
upon  a  single  presented  instance  is  sufficient  for  the 
establishment  of  a  universal  extending  in  range  over 
imagery  as  well  as  impression. 

§  5.  Passing  now  to  other  experiential  judgments, 
which  are  not  merely  sensational,  we  may  illustrate 
intuitive  induction  from  introspective  judgments. 
For  instance,  when  I  judge  that  it  is  the  pleasure  of 
this  or  that  experience  which  causes  me  to  desire  it, 
I  am  implicitly  universalising  and  maintaining  that  the 


J.  L.11 


13 


194 


CHAPTER  VIII 


^ 


INTUITIVE  INDUCTION 


195 


pleasure  of  any  experience  would  cause  me  to  desire 
it.    And  again,  when  I  judge  that  the  greater  resultant 
desire  for  one  possible  alternative  than  for  any  other 
causes  me  to  will  that  alternative,  I  am  judging  that 
this  will  hold   for  all  my  volitional  experiences.    An 
important  sub-class  of  experiential  judgments  which 
are  intuitively  inductive  consists  of  moral  judgments. 
Thus,  when  anyone  judges  that  a  certain  act  charac- 
terised with  a  sufficient  degree  of  precision  is  cowardly, 
or  dishonest,  or  generous,  he  is  implicitly  judging  that 
all  acts  of  the  same  specific  character  would  be  charac- 
terisable  by  the  corresponding  moral  attribute.    That 
this  is  not  a  case  of  mere  abstraction  is  clear  when  we 
consider  that  the  characteristics  used   to   define  the 
nature  of  the  action  are  other  than  ethical,  and  that 
the  judgment  is  therefore   synthetic.    This   intuitive 
aspect  of  moral  judgments  assumes  importance  as  re- 
conciling the  two  forms  of  ethical  intuitionism  to  which 
Sidgwick  refers  as  Perceptual  and  Dogmatic,  the  first 
of  which  stands  for  the  particular,  and  the  second  for 
the  universal,  intuition.    For,  in  my  view,  the  Dogmatic 
form  of  intuition  is  not  genuinely  intuitive  except  so 
far  as  it  is  based  on  the  Perceptual.    Instead,  therefore, 
of  distinguishing  moralists  according  to  what  they  hold 
to  be  the  nature  of  an  ethical  intuition,  it  is  more  im- 
portant  to   distinguish  them  according  as  they  base 
their  doctrine  upon  genuinely  intuitive  judgments,  e.g. 
Kant;   or  upon  judgments  accepted  on  authority  as 
expressions  of  the  voice  of  God,  e.g.  Butler. 

§  6.  The  gulf  between  experiential  and  formal 
intuition  is  bridged  by  considering  certain  intermediary 
forms  of  intuitive  apprehension  in  which,  according  as 


) 


the  range  of  universality  increases,  we  depart  further 
from  the  merely  experiential  and  approach  nearer  to 
the  merely  formal  type.  A  typical  case  is  the  merely 
experiential  judgment  that  red  and  green  cannot  both 
be  predicated  of  the  same  visual  area  by  one  person 
at  one  time.  The  judgment  is  first  universalised  when 
the  experient  sees  that  the  same  holds  of  all  cases  of 
the  specific  determinates  red  and  green.  But  this  judg- 
ment almost  immediately  passes  into  the  wider  universal 
that  any  two  different  determinates  under  the  deter- 
minable colour  are  similarly  incompatible.  And  when 
lastly  the  experient  extends  the  range  of  his  judgment 
to  all  determinables,  he  has  reached  a  formal  intuition, 
namely  that  any  two  different  determinates  under  any 
determinable  are  incompatible. 

To  this  formal  type  of  intuition  belong  all  intuitively 
apprehended  mathematical,  as  well  as  purely  logical, 
formulae.  For  instance,  the  algebraical  formula  known 
as  the  Distributive  Law  is  intuitively  reached  in  some 
such  way  as  this :  perceiving  that 

3  times  2  ft.  -I-  3  times  5  ft.  =  3  times  (2  ft.  -h  5  ft.) 

we  immediately  realise  that 

4 times  7  days  +  4 times  9 days  =  4  times  (7  days  +9days), 

and  in  this  step  we  are  virtually  apprehending  the  Dis- 
tributive Law  symbolically  expressed  thus : 

n  times  P-^-n  times  Q  =  n  times  (P-\-  Q) 

where  n  stands  for  any  number,  and  P  and  Q  for  any 
two  homogeneous  quantities. 

A  logical  example  of  a  similar  nature  is  the  formula 
of  the  simple  conversion  of  particular  affirmative  pro- 
positions.   This  is  reached  by  perceiving,  for  instance, 

13—2 


196 


CHAPTER  VIII 


that  'Some  Mongols  are  Europeans'  would  imply  that 
'Some  Europeans  are  Mongols,'  and  at  the  same  time 
that 

*Some  beings  incapable  of  speech  have  the  same 
degree  of  intelligence  as  men'  would  imply  that  *Some 
beings  having  the  same  degree  of  intelligence  as  men 
are  incapable  of  speech.' 

This  leads  to  the  virtual  apprehension  of  the  universally 

expressed  implication: 

'Some  things  that  are  /  are  ^'  would  imply  that 
'Some  things  that  are  q  are/' 

where  /  and  q  stand  for  any  adjective. 

§  7.  This  example  of  the  establishment  of  logical 
formulae  by  means  of  intuitive  induction  has  an  educa- 
tional importance  in  correcting  a  certain  prevalent  con- 
ception of  the  function  of  logic.  What  is  called  formal 
or  deductive  logic  is  usually  taught  by  first  presenting 
general  principles  in  a  more  or  less  dogmatic  form,  with 
the  result  that  the  learner  is  apt  to  use  these  principles 
merely  as  rules  to  be  applied  mechanically  in  testing 
the  validity  of  logical  processes.  Instead  of  leading  him 
to  conceive  of  these  rules  as  externally  imposed  impera- 
tives, an  appeal  should  be  made  to  him  to  justify  all 
fundamental  principles  by  the  exercise  of  his  own 
reasoning  powers;  and  this  exercise  of  power  will  in- 
volve the  process  of  intuitive  induction. 


\ 


I 


CHAPTER  IX 

SUMMARY  (INCLUDING  GEOMETRICAL)  INDUCTION 

§  I.  The  term  summary  induction  is  here  chosen 
in  preference  to  what,  in  the  phraseology  of  the  old 
logicians,  was  called  'perfect  induction,'  to  denote  a 
process  which  Mill  regarded  as  not  properly  to  be  called 
induction ;  on  the  ground  that  the  conclusion  does  not 
apply  to  any  instances  beyond  those  constituting  the 
premiss.  Mill's  contention  can  certainly  be  justified  in- 
asmuch as  the  process  involves  precisely  the  same 
logical  principles,  and  these  alone,  that  govern  ordinary 
deduction.  In  fact,  the  process  of  summary  induction 
may  be  expressed  in  the  form  of  a  syllogism  in  the  first 
figure.    For  example : 

Major  Premiss,  '  Sense  and  Sensibility'  and  '  Pride 
and  Prejudice'  and  '  Northanger  Abbey'  and  ' Mansfield 
Park'  and  'Emma'  and  'Persuasion'  deal  with  the 
English  upper  middle  classes. 

Minor  Premiss,  Every  novel  of  Jane  Austen  is 
identical  either  with  'Sense  and  Sensibility'  or  with 
'Pride  and  Prejudice '  or  with '  Northanger  Abbey ' or  with 
'Mansfield  Park'  or  with  'Emma'  or  with  'Persuasion/ 

Conclusion,  .',  Every  novel  of  Jane  Austen  deals 
with  the  English  upper  middle  classes. 

Here  the  enumeration  standing  as  subject  in  the  major 
premiss  is  the  same  as  the  enumeration  standing  as 
predicate  in  the  minor  premiss.  But,  in  the  former, 
reference  is  made  to  ever}^  one  of  the  collection,  in  the 


y 


:\ 


198 


CHAPTER  IX 


latter  to  some  one  or  other.  This  precisely  corresponds 
to  the  characteristic  of  first  figure  syllogism ;  namely 
that  the  middle  term  is  distributed  as  subject  of  the 
major  and  undistributed  as  predicate  of  the  minor.  In 
text-book  illustrations  of  perfect  induction  the  minor 
premiss  is  almost  invariably  omitted,  because  the  illus- 
trations chosen — such  as  the  Apostles  or  the  months  of 
the  year — are  so  familiar  that  the  completeness  of  the 
enumeration  is  assumed  to  be  known  by  every  ordinary 
reader  and  therefore  does  not  require  to  be  expressed 
in  a  separate  minor  premiss.  The  same  process  is 
exhibited  by  an  example  in  which  each  of  the  items 
enumerated  is  a  universal  instead  of  being  a  singular : 

Every  parabola  and  every  ellipse  and  every  hyper- 
bola meet  a  straight  line  in  less  than  3  points. 

Every  conic  section  is  either  a  parabola  or  an  ellipse 
or  a  hyperbola. 

.-.  Every  conic  section  meets  a  straight  line  in  less 
than  3  points. 

§  2.  Another  case  of  perfect  induction,  which  has 
specific  bearing  upon  induction  in  general,  may  be 
expressed  symbolically  in  the  following  syllogism: 

Sj  and  i"2  ...  and  s^  are/. 

Every  examined  case  of  m  is  identical  either  with 

s^  or  with  s^  ...  or  with  s^- 
.•.  Every  examined  case  of  ;;^  is/. 

A  summary  or  perfect  induction  of  this  form  is  the 
necessary  preparatory  stage  in  gathering  together  the 
relevant  instances  for  establishing  an  unlimited  generali- 
sation. For  the  conclusion  thus  obtained,  constitutes  the 
premiss  from  which  we  directly  infer,  with  a  higher  or 
lower  degree  of  probability,  that  *  Every  case  of  w  is/. 


SUMMARY  INDUCTION 


199 


Whewell  pointed  out  the  importance  and  difficulty  of 
discovering  '  the  concept  /  under  which  the  instances 
are  colligated.'    He,  in  agreement  with  other  critics  of 
Mill,  accordingly  held  that  the  process  of  induction  was 
completed  in  the  discovery  of  this  colligating  concept, 
on  the  ground  that  this  process  alone  required  some- 
thing like  genius  to  perform,  while  it  is  the  easiest 
thing  in  the  world  to  pass  from  every  examined  instance 
to  every  instance.    Mill,  on  the  other  hand,  considered 
that  this  process  only  supplied  the  requisite  premiss  for 
a  genuine  inductive  inference.    To  illustrate  his  view, 
Whewell  had  chosen  Kepler  s  famous  discovery  of  the 
formula  for  the  orbit  of  the  planets,  and  it  was  towards 
this  illustration  that  Mill  directed  his  criticism.    Ex- 
pressed in  terms  of  the  above  used  symbols. 

Let  m  stand  for  'positions  of  a  certain  moving  planet/ 
5i,  5^  ...  ^n    „      „    *  the  several  observed  positions,' 
and       /     „      „   *  being  a  point  on  a  certain  ellipse.' 
The  syllogism  which  expresses  the  process  of  perfect 
induction  used  by  Kepler  will  then  be  as  follows: 

*Each  of  several  observed  positions  is  a  point  on 
a  certain  ellipse. 

'  Every  examined  position  of  a  certain  moving  planet 
is  identical  either  with  one  or  with  another  of  these 
several  observed  positions.' 

.-.  *  Every  examined  position  of  the  moving  planet 
is  a  point  on  that  ellipse.' 

This  formula  had  not  been  discovered  by  any  previous 
astronomer,  and,  on  the  grounds  already  assigned, 
Whewell  maintained  that  the  discovery  constituted  the 
completion  of  the  induction  To  this  Mill  demurred, 
because  by  induction  he  meant  a  process  in  which  the 


200 


CHAPTER  IX 


conclusion  is  an  unlimited  universal  extending  beyond 
examined  instances ;  he,  however,  failed  to  observe  that 
Kepler  had  actually  gone  beyond  the  examined  instances 
and  had  described  the  complete  orbit  of  the  planet  by 
inferring  that  what  held  of  the  examined  positions 
would  hold  of  all  the  interpolated  positions.  Kepler 
had  thus  unconsciously  made  a  genuine  induction  in  the 
sense  required  by  Mill.  Whewell  was  concerned  with 
the  art  of  discovery,  and  therefore  held  that  the  essen- 
tial factor  in  induction  was  the  discovery  of  the  colli- 
gating concept;  whereas  Mill  was  concerned  with  the 
science  of  proof,  and  therefore  held  that  the  essential 
factor  in  any  induction  (that  was  not  merely  formal  or 
demonstrative)  was  the  inferential  extension  from  ex- 
amined to  unexamined  instances. 

§  3.  Having  illustrated  the  process  of  summary  (or 
perfect)  induction  by  familiar  examples,  in  which  the 
conclusion  applies  to  a  finite  number  of  cases  which  are 
enumerable,  we  proceed  to  consider  a  more  interesting 
type  of  summary  induction  in  which  the  conclusion 
applies  to  an  infinite  number  of  cases  which  are  non- 
enumerable.  This  type  occurs  in  geometrical  proofs  of 
geometrical  theorems,  and  has  been  more  or  less  con- 
fused on  the  one  hand  with  merely  intuitive,  and  on  the 
other  hand  with  problematic  induction.  It  differs,  how- 
ever, from  the  former  in  that  its  conclusion  cannot  be 
reached  from  an  examination  of  one  or  of  a  few  instances; 
and  from  the  latter  in  that  the  conclusion  does  not 
extend  beyond  the  range  of  the  examined  instances — 
these  being  apprehended  in  their  infinite  totality. 

It  is  well  known  that  there  are  two  modes  by  which 
geometrical  theorems  may  be  proved,  viz.  'analytical' 


I 


( 


,^- 


SUMMARY  INDUCTION 


201 


and  'geometrical.'  Strict  analytical  proof  has  the  same 
logical  character  as  algebraical  proof,  and  comes  under 
the  head  of  functional  deduction.  Such  proofs  do  not 
require  the  aid  of  geometrical  figures.  But  the  geo- 
metrical method  of  proof  depends  essentially  upon  the 
use  of  such  figures.  It  may  further  be  pointed  out  that 
the  analytical  method  has  an  indefinitely  wider  scope 
than  the  geometrical.  For  example,  by  employing  mere 
analysis  we  can  construct  spaces  of  various  different 
forms  other  than  Euclidian;  and  certainly  a  geometrical 
method  would  be  impossible  except  as  applied  to  our 
space  which  is  presumed  to  be  Euclidian.  The  actual 
procedure  in  constructing  any  non-Euclidian  space  is  to 
bring  forward  some  four  or  five  axioms  which  must  be 
{a)  independent  of  one  another,  and  (^)  mutually  con- 
sistent. These  axioms,  however,  are  not  put  forward 
categorically,  but  purely  hypothetically ;  it  follows,  there- 
fore, that  the  theorems  which,  for  convenience  are  said 
to  be  deduced  from  the  axioms,  should  be  more  strictly 
said  to  be  implied  by  the  axioms.  Such  systems,  there- 
fore, are  throughout  implicative  and  not  inferential.  In 
other  words,  a  supposed  space,  definable  by  any  chosen 
set  of  axioms,  would  have  such  and  such  other  charac- 
teristics which  these  axioms  would  formally  imply.  On 
the  other  hand,  the  geometrical  method  is  a  method  of 
proof  or  inference,  inasmuch  as  we  accept  its  conclusions 
as  true  only  because  we  have  accepted  its  axioms  as 

true. 

§  4.  We  must  therefore  examine  the  process  by 
which  the  axioms  of  geometry  are  established.  These, 
qua  axioms,  are  not  reached  by  deduction ;  and,  since 
they  are  universal  in  the  specific  sense  that  they  apply 


202 


CHAPTER  IX 


; 


SUMMARY  INDUCTION 


203 


for  an  infinite  number  of  possible  instances,  it  would 
seem  that  some  form  of  induction  is  required  for  their 
establishment ;  unless  we  adopt  the  view  that  they  are 
obtained  by  a  process  which  embraces  all  possible  cases 
in  a  single  act  of  direct  intuition.  This  latter  appears 
to  be  the  view  of  Kant  who  held,  as  regards  geometry, 
that  our  intuitions  are  from  the  first  universal,  and  that 
they  therefore  function  as  premisses  for  deducing  any, 
or  any  other,  given  case. 

In  order  to  examine  this  question  let  us  take  the 
familiar  axiom  conveniently  expressed   in   the  form: 
*  Two  straight  lines  terminating  at  the  same  point  cannot 
intersect  at  any  other  point.'  This  is  the  most  important 
axiom  which  does  not  hold  of  non-Euclidian  spaces  in 
general.    Independently,  however,  of  the  nature  of  any 
other  kind  of  space,  the  axiom  certainly  represents  the 
manner  in  which  we  actually  intuite  our  space,  whether 
falsely  or  truly.    Now  this  axiom,  in  its  universality, 
can  be  established  only  by  means  of  imagery  and  not 
by  mere  perception;  for  the  compass  over  which  the 
axiom  holds  is  beyond  the  range  of  actual  perception. 
For  in  the  first  place  it  is  only  through  imagery  that  we 
can  represent  a  line  starting  from  a  certain  point  and 
extending  indefinitely  in  a  certain  direction ;  and,  in  the 
second  place,  we  cannot  represent  in  perception  the 
infinite  number  of  different  inclinations  or  angles  that 
a  revolving  straight  line  may  make  with  a  given  fixed 
straight  line.    We  may,   however,   by  a  rapid  act  of 
ocular  movement  represent  a  line  revolving  through 
360°  from  any  one  direction  to  which  it  returns.    In  this 
imaginative  representation  the  entire  range  of  variation, 
covering  an  infinite  number  of  values,  can  be  exhaus- 


A 


\  i\ 


tively  visualised  because  of  the  continuity  that  charac- 
terises the  movement.  It  is  only  if  such  a  process  of 
imagery  is  possible  that  we  can  say  that  the  axiom  in 
its  universality  presents  to  us  a  self-evident  truth.  It 
is  therefore  this  species  of  summary  induction  that  is 
employed  to  establish  geometrical  axioms — differing,  as 
explained  above,  on  the  one  hand  from  mere  intuitive 
induction,  inasmuch  as  one  or  a  few  specific  cases  would 
not  constitute  an  adequate  premiss ;  and,  on  the  other 
hand,  from  induction  in  Mill's  specific  sense,  since  the 
conclusion  does  not  go  beyond  the  premisses  taken  in 
their  totality. 

§  5.  I  shall  further  maintain  that  if,  in  the  course  of 
a  geometrical  proof  which  may  involve  several  succes- 
sive steps,  the  perception  or  image  of  a  figure  is  required 
for  any  single  step,  this  is  because  we  have  to  go 
through  precisely  the  same  process  of  summary  induc- 
tion, embracing  an  infinite  number  of  specialised  cases 
of  which  the  figure  under  inspection  is  one — all  of  these 
being  included  in  the  subject  of  the  universal  conclusion 
to  be  proved  at  that  step.  Speaking  generally,  in  any 
one  demonstrative  step,  the  major  premiss  is  a  universal 
previously  established,  and  from  this  universal  major  it 
is  required  to  establish  a  new  universal  conclusion.  It 
is  obvious  that  this  can  only  be  done  by  means  of  a 
universal  minor;  and  it  is  in  the  establishment  of  the 
universality  of  the  minor  that  consists  the  logical  func- 
tion of  the  figure.  The  arbitrarily  chosen  figure  under 
inspection  can  only  be  used  as  a  minor  term  to  prove 
the  conclusion  about  that  single  figure;  and  hence,  to 
obtain  the  required  universal  conclusion,  the  minor 
must  be  universalised  by  the  same  logical  process  that 


204 


CHAPTER  IX 


is  used  for  establishing  the  explicit  axioms.  Now  the 
Euclidian  geometry  might  have  been  established  by 
purely  analytical  methods ;  provided  first,  that  a  sufficient 
number  of  axioms  had  been  explicitly  formulated ;  and 
secondly  that  each  of  these  axioms  had  been  established 
for  itself  by  the  process  of  summary  induction.  Such 
an  analytical  system  would  dispense  with  the  use  of 
figures  as  objects  either  of  perception  or  of  imagery  in 
the  course  of  the  proof ,  these  being  only  required  in  the 
process  of  establishing  the  axioms  themselves. 

To  show  by  specific  illustration  how  the  geometrical 
proof  uses  a  figure,  we  will  select  a  very  frequently 
assumed,  but  not  explicitly  stated  axiom,  which,  in 
Euclid's  proofs  is  required  to  supplement  the  explicit 
axiom  *the  whole  is  greater  than  its  part,'  or  more 
precisely,  *the  v/hole  is  equal  to  the  sum  of  its  parts.' 
Before  this  explicit  axiom  can  be  used,  we  must  be 
satisfied  that  the  two  elements  of  the  figure,  one  of 
which  is  to  be  greater  than  the  other,  do  stand  in  the 
relation  of  whole  to  part.  The  axiom  to  which  I  refer 
is  actually  employed  by  Euclid  and  most  geometricians 
in  the  propositions  numbered  5,  6,  7,  16,  18,  20,  21,  24, 
and  26  in  Euclid,  Book  I.    It  may  be  formulated  as 

follows :  The  angle  sub- 
tended at  any  point  by  a 
part  of  a  line  is  part  of  the 
angle  subtended  by  the 
whole  line.'  If  the  reader 
is  not  familiar  with  this 
new  axiom,  he  must  go 
through  a  process  in  which  he  imagines  a  line  revolving 
in  a  plane  through  a  point  (G)  from  some  initial  direc- 


\ 


A'f 


4, 
I 


*  i 


r 


,^ 


■:i 


n  »■  *V  ( 


I 


SUMMARY  INDUCTION 


205 


tion  (OA)  to  a  final  direction  (OC),  so  that  it  will 
intersect  the  whole  line  {AC)  in  a  series  of  successive 
points.  In  this  way,  and  in  this  way  alone,  can  he 
accept  the  universality  of  the  required  conclusion  that 
the  angle  AOC  is  greater  than  the  angle  A  OB.  In 
Euclid's  theorems  enumerated  above  it  will  be  found 
that  this  axiom  is  required  in  every  case  to  establish 
the  conclusion  that  a  certain  angle  is  greater  than 
another  ;  and  that  this  conclusion  is  a  necessary  step  in 
the  further  progress  of  each  proof. 

Geometrical  induction  involves,  in  addition  to  the 
summary  process  above  explained,  two  further  pro- 
cesses which  are  of  the  nature  of  intuitive  induction, 
as  explained  in  the  preceding  chapter.  Of  these  two, 
the  first  is  concerned  with  absolute  position,  the  second 
with  absolute  magnitude.  Thus,  having  reached  a 
universal  by  summary  induction  limited  to  figures  oc- 
cupying a  certain  position,  it  is  by  intuitive  induction 
that  we  pass  to  figures  of  the  same  specific  shape  and 
magnitude  occupying  any  other  possible  position ;  and 
again  from  a  figure  imaged  as  having  a  certain  magni- 
tude, to  figures  of  the  same  specific  shape  but  of  any 
other  possible  magnitude.  I  have  described  these  two 
processes  as  of  the  nature  of  intuitive  induction,  in 
which  we  universalise  by  abstracting  from  variable 
position  and  from  variable  magnitude  ;  but  they  might 
otherwise  be  regarded  as  involving  the  conception  of 
position  and  magnitude  as  being — not  absolute — but 
relative  to  the  percipient's  own  position  and  to  his  dis- 
tance from  the  figure  depicted  in  imagination. 

§  6.    Having  illustrated  the  proper  use  of  the  geo- 
metrical figure,  we  shall  proceed  to  illustrate  what  may 


) 


206 


CHAPTER  IX 


SUMMARY  INDUCTION 


be  called  its  abuse  ;  and  give,  by  means  of  a  figure,  an 
alleged  proof  that  every  triangle  is  isosceles  : 

To  prove  that  every  Triangle  is  Isosceles, 

Let  the  bisector  of 
the  vertical  angle  A 
meet  the  perpendicular 
bisector  of  the  base^C, 
whose  middle  point  is 
D,  at  the  point  O,  Join 
BO,  CO,  and  draw  OE 
perpendicular  to  AC, 
and  OF  perpendicular 
XoAB,    Then, 

(i)  the  triangles  BOD,  COD,  are  congruent;  for 
BD  —  CD\  OD  is  common;  and 
LBDO^  L  CDO] 
.'.  BO  =  CO. 

(2)  the  triangles  A  OB,  A  OB,  are  congruent;  for 
AO  is  common;   L  OAB=  L  OAF\  and 

LOEA^  LOFA] 
.-.  AB  =  AF2ind  OE^OF, 

(3)  the  triangles  COE,  BOB  are  congruent;  for, 
by  (i)  CO  =  BO',  and  by  (2)  OE^OB-,  and 
hence  CO"  -  OE^  =  BC^  -  0B^\ 

i.e.  (since  CEO  and  BBO  are  rt  L  )  CE^^BB^. 
Hence,  by  (2)  AE^AB^xid.,  by  (3),  CE^BB; 
.-.  by  addition  AC=AB.     q.e.d. 

Here  we  see  that  the  axiom  :  *the  whole  is  greater 
than  its  part'  is  used  in  its  more  precise  form,  'the 
whole  is  equal  to  the  sum  of  its  parts.'  Now  before  we 
can  state  as  regards  the  straight  line  ABB,  that 

AB  =  AB+BB, 

we  must  be  sure  that  AB,  BB  are  really  parts  of  AB; 


I 


>  f\ 


*^'  u 


207 


whereas  if  B  was  beyond  AB,  then  AB  would  be  the 
whole  and  AB,  BBv^ouXd  be  its  parts. 

The  fallacy  incurred  in  this  proof  arises  from  the 
mistaken  intuition  that  the  bisector  of  the  vertical  angle 
A  meets  the  perpendicular  bisector  of  the  base  BC  at 
a  point  O  inside  the  triangle.  By  drawing  an  incorrect 
figure  and  thus  convincing  ourselves  of  the  false  con- 
clusion, we  had  unconsciously  universalised  from  the 
figure  before  us  that  for  every  case  the  two  bisectors 
would  meet  at  a  point  within  the  triangle,  this  being  in- 
dicated in  the  figure  as  drawn.  In  other  words,  we  have 
swallowed  the  relation  presented  in  the  drawn  figure 
as  being  universalisable,  without  having  gone  through 
the  necessary  summary  induction. 

We  may  proceed  to  draw  the  corrected  figure. 
From  this  we  reach,  as  before,  the  two  conclusions 

AE^AB^ViA  CE  =  BB. 


But  now  we  see  that 

AC=AE  +  EC, 
while  AB^AB-BB. 


208 


CHAPTER  IX 


SUMMARY  INDUCTION 


209 


Euclidian  demonstration  professes  to  be  based  on 
pure  reasoning,  in  such  a  'manner  that  the  figure  may 
be  drawn  quite  inaccurately,  and  yet  the  force  of  the 
proof  be  equally  cogent.  But  it  may  happen,  as  in  the 
case  before  us,  that  the  figure  is  drawn  with  a  degree 
of  inaccuracy  which  affects  the  proof;  because  the 
particular  demonstration,  involving  unconscious  refer- 
ence to  the  figure  drawn,  has  been  illegitimately  uni- 
versalised. 

§  7.  My  explanation  of  the  logical  function  of  the 
figure  in  geometrical  demonstration  differs  fundamen- 
tally from  that  put  forward  by  Mill,  who  maintains  that 
it  is  by  parity  of  reasoning  that  what  is  apprehended 
to  be  true  for  the  one  drawn  figure,  is  apprehended  to 
be  true  for  any  other  figure  (within  the  scope  of  the 
conclusion).  But  the  passage  from  the  demonstration 
for  one  case  to  that  for  any  other  case  can  only  be  said 
to  exhibit  *  parity  of  reasoning '  when  the  two  demon- 
strations have  the  sdim^  form.  Taking  for  example  the 
two  demonstrations : 

(i)  every  misp)  this  S  is  m\  therefore  this  S  isp; 
(2)  every  m  is/;  that  S  is  m-,  therefore  that  Sisp; 

we  may  certainly  pass  by  parity  of  reasoning  from  (i) 
to  (2)  inasmuch  as  both  arguments  are  of  the  same 
form,  the  words  *this'  and  'that'  indicating  difference 
in  matter.  In  ascribing  the  same  form  to  (i)  and  (2), 
what  is  meant  is  that  the  relation  of  implication  between 
the  premisses  and  conclusion  of  the  one  is  the  same  as 
that  between  the  premisses  and  conclusion  of  the  other. 
But  in  order  to  use  an  implication  for  the  purposes  of 
inference,  we  should  have  to  assert  *  This  S  is  m'  for 


i 


/ 


case  (i)  and  *That  S  is  m'  for  case  (2);  for  although 
the  relation  of  implication  is  the  same  in  the  two  argu- 
ments, it  does  not  follow  from  having  asserted  the 
minor  of  the  one  that  we  can,  on  this  ground,  assert 
the  minor  of  the  other.  Now,  in  order  to  establish 
the  required  conclusion  *  Every  S  is /'  we  must  first 
establish  the  universalised  minor  'Every  6*  is  m'  No 
reasoning  process  (in  the  accepted  meaning  of  the 
term)  would  enable  us  to  pass  from  the  case  'This  6*  is 
m'  to  'That  5  is  m'  and  to  'That  other  5  is  m'  ad 
infinitum ;  and  the  only  mode  of  establishing  the  re- 
quired universal  minor  'Every  5*  is  w'  is  through  some 
process  of  induction,  the  nature  of  which  we  have  been 
describing. 


J.  L.n 


14 


CHAPTER  X 

DEMONSTRATIVE  INDUCTION 

§  I.  Having  so  far  examined  intuitive  and  summary 
induction,  we  now  pass  to  the  third  type  of  inductive 
inference  distinguished  at  the  outset,  namely  demon- 
strative induction.  As  its  name  suggests,  this  form  of 
inference  partakes  both  of  the  nature  of  demonstration 
and  of  induction.  It  includes  several  different  forms, 
the  characteristics  common  to  them  all  being  (i)  that 
they  are  demonstrative,  in  the  sense  that  the  conclusion 
follows  necessarily  from  the  premisses;  and  (2)  that 
they  are  inductive,  in  the  sense  that  the  conclusion  is 
a  generalisation  of  a  certain  premiss  or  set  of  premisses 
which,  taken  as  a  collective  whole,  may  be  spoken  of 
as  'the  instantial  premiss.'  The  possibility  of  arriving 
demonstratively  at  aconclusion  wider  than  the  premisses, 
depends  here  upon  the  nature  of  the  major  premiss, 
which  is  not  only  universal  but  composite.  In  short 
demonstrative  induction  may  be  described  as  that  form 
of  inference  in  which  one  premiss  is  composite  and  the 
other  instantial ;  the  conclusion  being  a  specification  of 
the  former  and  a  generalisation  of  the  latter. 

§  2.  In  explaining  the  nature  of  demonstrative  in- 
duction as  above  described,  the  composite  nature  of  the 
major  premiss  brings  us  back  to  those  fundamental 
modes  of  inference  specified  in  Part  I,  Chapter  III  on 
compound  propositions.  There  P  and  Q  are  taken  to 
stand  for  any  propositions,  and  four  composite  relations 


I 


5 


4» 


% 


J 


DEMONSTRATIVE  INDUCTION 


211 


are  distinguished  in  which  P  may  stand  to  Q\  (a)  Im- 
plicative, leading  to  the  Ponendo  Ponens]  {b)  Counter- 
implicative,  leading  to  the  Tollendo  Tollens ;  {c)  Alter- 
native, leading  to  the  Tollendo  Ponens  \  {d)  Disjunctive, 
leading  to  the  Ponendo  Tollens: 

{a)  UP  then  Q,  but  P]  therefore  Q. 

(d)  U  Q  then  P,     but  not  P\  therefore  not  Q, 

(c)  Either  P  or  Q,  but  not  P\  therefore  Q, 

(d)  Not  both /^  and  Q,  hutP;  therefore  not  Q. 

In  these  composite  premisses,  we  shall  take  the  impli- 
cates and  alternants  to  stand  for  universal  propositions, 
and  the  implicants  and  disjuncts  to  stand  for  particular 
propositions.  This  secures,  for  each  case,  a  form  of 
inference  in  which  a  particular  or  singular  premiss  yields 
a  universal  conclusion.    Thus: 

{a)    If  some  6"  is/,  then  every  7"is  ^; 
but  this  S  IS  py 
.'.  every  7"  is  q, 

(6)    If  some  7"  is  ^,  then  every  5  is/; 
but  this  S  isp\ 
.*.  no  7" is  q. 

(c)  Either  every  5  is/,  or  every  Tis  q; 

but  this  5  is/', 
.  •.  every  T  is  q. 

(d)  It  cannot  be  that  some  5  is/,  and  some  7"  is  ^; 

but  this  S  is/, 
.  •.  no  7"  is  ^. 

In  the  above  formulae,  it  will  be  observed  that  the 
simple  or  categorical  premiss  is  not  the  precise  equiva- 
lent or  contradictory,  as  the  case  may  be,  of  the  corre- 
sponding proposition  that  occurs  in  the  composite 
premiss;  for  'this  5  is  /'   is  more  determinate  than 

14—2 


212 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


213 


'some  5  is  //  being  one  of  its  superimplicants ;  and 
again  'this  Sisp"  is  not  the  mere  contradictory  of 'every 
5  is/,'  being  one  of  its  contraries  or  superopponents. 
The  categorical  premiss  having  been  in  this  way 
strengthened,  the  conditions  of  vaUd  inference  are  still 
satisfied.  In  short,  we  have  taken  as  our  instantial 
premiss  a  specific  instance  characterised  determinately. 
The  object  of  this  is  to  illustrate  the  symbolic  formulae 
by  concrete  examples  which,  when  further  developed, 
will  exhibit  the  nature  of  demonstrative  induction  in  its 
most  important  scientific  forms.  Consider  the  following 
illustrations  of  the  symbolic  formulae : 

[a)  If  some  one  recordec  miracle  has  been  shown 
to  have  happened,  then  every  natural  phenomenon  has 
a  supernatural  factor;  but  such  or  such  recorded  miracle 
has  been  shown  to  have  happened;  therefore  every 
natural  phenomenon  has  a  supernatural  factor. 

{b)  If  some  one  female  member  of  a  Board  had 
lowered  the  educational  standard  in  her  university, 
every  woman  would  have  submitted  to  exclusion  from 
the  Cambridge  Senate;  but  Miss  C.  has  not  submitted 
to  exclusion  from  the  Cambridge  Senate;  therefore  no 
female  member  of  a  Board  has  lowered  the  educational 
standard  in  her  university. 

(c)  Either  every  Protectionist  country  is  financially 
handicapped  or  every  economist  of  the  old  school  is 
mistaken;  but  America  is  commercially  prosperous; 
therefore  every  economist  of  the  old  school  was  mis- 
taken. 

{d)  It  cannot  be  that  some  variations  can  be  arti- 
ficially produced  in  domesticated  animals,  while  there 
are  some  species  whose  characters  are  unaffected  by 
their  environment ;  but  some  variations  have  been  arti- 
ficially produced  in  the  pigeon;  therefore  there  are  no 


r 


species  whose  characters  are  unaffected  by  their  environ- 
ment. 

These  illustrations  would  be  regarded  by  those  logicians 
who  divide  all  inferences  into  inductive  and  deductive, 
as  being  of  the  nature  of  deduction  rather  than  of  in- 
duction, because  the  universal  conclusion  is  not  a 
generalisation  of  the  instantial  premiss.  In  contrast  to 
these  we  will  therefore  now  select  a  set  which  will  be 
recognised  as  of  the  nature  of  induction ;  inasmuch  as 
here  the  universal  conclusion  in  each  case  is  a  generali- 
sation of  the  instantial  premiss.  These  new  examples 
are  applications  of  the  same  symbolic  formulae  as  the 
preceding  set ;  they  differ  only  in  that  the  symbols  5 
and  T  will  now  stand  for  the  same  class,  whereas  in  the 
first  set  they  stood  for  different  classes. 

(a)  If  some  boy  in  the  school  sends  up  a  good 
answer,  then  all  the  boys  will  have  been  well  taught ; 
the  boy  Smith  has  sent  up  a  good  answer;  therefore 
all  the  boys  have  been  well  taught. 

{b)  If  a  single  authoritative  person  had  witnessed 
the  alleged  occurrence,  then  everyone  would  have  be- 
lieved it ;  but  Mr  S.  is  incredulous ;  therefore  no 
authoritative  person  could  have  witnessed  the  occur- 
rence. 

{c)  Either  every  act  of  volition  is  determined  or 
every  act  of  volition  is  free;  but  by  introspection  I  am 
sure  that  a  certain  act  of  mine  was  undetermined ;  there- 
fore every  volition  is  free. 

{d)  It  is  impossible  to  suppose  that  any  modern 
theologians  are  genuine  scholars  while  others  have 
remained  orthodox;  Dean  I.  is  a  genuine  scholar; 
therefore  no  modern  theologian  could  have  remained 
orthodox. 


/. 


214 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


215 


§  3.  Returning  to  the  symbolically  expressed  formu- 
lae, and  substituting  /  or  p,  as  the  case  may  be,  for  q, 
as  well  as  5^  for  Z",  the  composite  premisses  will  assume 
the  following  still  more  specialised  form : 

{a)    If  some  S  is  p  then  every  5*  is/. 
(d)    If  some  S  isp  then  every  S  is/. 

(c)  Either  every  5  is/  or  every  S  is/. 

(d)  Not  both  some  6*  is/  and  some  S  is/. 

It  will  be  seen  that  these  four  composite  premisses  are 
formally  equivalent  to  one  another,  and  that  by  adding 
the  categorical  premiss  '  This  6"  is  / '  we  may  conclude 
in  each  case  that  '  Every  S  is/.'  Now  we  may  transform 
the  alternation  of  universals  in  (c)  and  the  disjunction 
of  particulars  in  (d)  by  substituting  for  /  and/  any  set 
of  predicates  p,  q,  r,  t,  v  ,.,  for  the  alternative  pro- 
position (c\  and  the  same  set  in  pairs  for  the  disjunctive 
proposition  {d),  thus  : 

(c)  Either  every  5  is/  or  every  5  is  ^  or  every  5 
is  r  or  . . .  etc. 

(d)  Not  both  'some  6*  is/  and  some  5  is  q'  and 
not  both  'some  S\sp  and  some  5*  is  r\..  etc.  etc. 

In  this  transformation  the  two  complexes  (c)  and  {d) 
are  no  longer  equivalents  but  rather  complementaries 
to  one  another.  If  the  categorical  premiss  'This  5*  is 
/'  is  now  introduced  we  may  infer  by  means  of  {d)  that 
'No  5  is  ^,'  'No  5  is  r/  'No  6*  is  t'  etc.,  so  that  all  but 
the  first  of  the  universal  alternants  in  (c)  is  rejected, 
and  again  the  universal  conclusion  'Every  5  is /*  is 
established.  The  need  of  combining  the  complemen- 
taries (c)  and  {d)  in  order  to  establish  the  required  uni- 
versal conclusion  is  apparent  when  we  consider  a  con- 


^ 


4 


¥ 


Crete  illustration.  In  the  following  example,  where  the 
predicates/,  q,  r ...  stand  respectively  for  'attacking 
the  Coalitionists/  'attacking  the  Liberals,'  'attacking 
the  Labour  Party'...  it  will  be  observed  that  the  com- 
posites {c)  and  {d)  retain  the  same  logical  force  as  in 
the  above  symbolisation,  although  somewhat  differently 
worded  • 

{c)  At  least  one  of  the  political  parties  was  attacked 
by  every  speaker  at  a  certain  sitting  of  the  Congress, 
and  {d)  not  more  than  one  of  the  parties  was  attacked 
at  that  sitting. 

Mr  X.  who  spoke  attacked  the  Coalitionist  Party. 
.-.  Every    speaker    at    the    sitting    attacked    the 
Coalitionist  Party. 

§  4.    Now,  if— instead  of/,  q,  r ...— we  take  deter- 
minates/, /',/"...  under  the  same  determinable  P, 
then  the  disjunctive  premiss  {d)  will  not  be  explicitly 
required,  because  it  is  accepted  a  priori  that  nothing 
can  be  characterised  by  both  of  any  two  determinates 
under  the  same  determinable.    What  remains  then  is 
the  universal  alternative  proposition    {c\  established, 
we  may  assume,   by  problematic   induction;   namely: 
*  Either  every  S  is  /,  or  every  S  is  /',  or  every  S  is/". . . 
running  through  all  the  determinates  under  P^  and 
this  may  be  summed  up  in  the  single  phrase  '  Every  5 
is  characterised  by  some  the  same  determinate  under  the 
determinable  P!    If  to  this  composite  premiss  is  added 
the  instantial  premiss  'This  5  is  /,'  the  universal  con- 
elusion   follows   that   'Every    5    is  /.'    This   trio   of 
propositions  represents  the  one  immediate  way  of  es- 
tablishing a  generalisation  demonstratively  from  a  single 
instance,  and  it  will  be  termed 


'•'! 


2l6 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


217 


The  Formula  of  Direct  Universalisation 

Composite  Premiss:  Every  5  is  characterised  by- 
some  the  same  determinate  under  the  determinable  P. 
Instantial  Premiss:  This  6"  is/. 
Conclusion:  .'.   Every  6*  is/. 

§  5.    To  take  a  typical  illustration  from  science : 

Every  specimen  of  argon  has  some  the  same  atomic 
weight. 

This  specimen  of  argon  has  atomic  weight  39*9. 
.'.  Every  specimen  of  argon  has  atomic  weight  39*9. 

In  this,  as  in  all  such  cases  of  scientific  demonstra- 
tion, the  major  premiss  is  established — not  directly,  by 
mere  enumeration  of  instances — but  rather  by  deductive 
application  of  a  wider  generalisation  which  has  been 
ultimately  so  established.  In  the  given  example  it  is 
assumed  that  a// the  chemical  properties  of  a  substance, 
defined  by  certain  'test'  properties,  will  be  the  same  for 
all  specimens ;  and  this  general  formula  is  applied  here 
to  the  specific  substance  argon,  and  to  the  specific  pro- 
perty atomic  weight.  The  assumption  in  this  case  is 
established  by  problematic  induction,  i.e.  directly  from 
an  accumulation  of  instances.  In  practically  all  experi- 
mental work,  a  single  instance  is  sufficient  to  establish 
a  universal  proposition :  when  instances  are  multiplied 
it  is  for  the  purpose  of  eliminating  errors  of  measure- 
ment. It  is  owing  to  the  fact  that  the  general  propo- 
sition, functioning  as  major  or  supreme  premiss,  has  the 
special  form  of  an  alternation  of  universals  that,  by 
means  of  a  minor  premiss  expressing  the  result  of  a 
single  observation,  we  are  enabled  to  establish  a  uni- 
versal conclusion.    This  conclusion,  in  accordance  with 


r 


t 

; 


•i 


<  H 


our  general  account  of  demonstrative  induction,  is  a 
specification  of  what  is  predicated  indeterminately  in 
the  universal  premiss,  and  a  generalisation  of  the  pro- 
position recording  the  result  of  a  single  observed 
instance. 

§  6.  The  most  important  extension  of  demonstrative 
induction  deals  with  such  methods  as  those  of  agree- 
ment and  difference  that  have  been  treated  by  Mill. 
We  propose  to  give  a  formal  account  of  methods  similar 
to  those  explained  by  Mill,  but  so  constructed  as  to 
render  them  strictly  demonstrative.  Many  critics  of 
MilFs  methods  have  treated  them  disparagingly  because 
of  his  failure  to  exhibit  their  formal  cogency;  while 
others  have  maintained  that  induction  should  not  profess 
to  exhibit  the  strictly  formal  character  that  is  ascribed 
to  syllogism  and  other  deductive  processes.  I  hold,  on 
the  contrary,  that  Mill's  methods  can  and  should  be 
exhibited  as  strictly  formal,  by  rendering  explicit  certain 
implicit  premisses  upon  which  the  cogency  of  the  argu- 
ment from  instances  in  any  given  case  depends;  and 
by  indicating  the  precise  conclusion  which  can  be  drawn 
from  the  instances  in  question.  The  implicit  premiss  is 
ultimately  established  by  a  process  of  problematic  in- 
duction, which  must  be  sharply  distinguished  from  the 
demonstrative  process  exemplified  by  the  methods. 
Mill's  exposition  differs  from  mine,  then,  in  three  pre- 
liminary respects.  In  the  first  place,  he  does  not  clearly 
distinguish  the  nature  of  direct  or  problematic  induction 
from  the  nature  of  the  process  conducted  in  accordance 
with  his  'methods  of  induction,'  which  he  appears  often 
to  regard  as  demonstrative.  This  confusion  is  particu- 
larly noticeable  when  we  contrast  his  different  modes 


i 


2l8 


CHAPTER  X 


of  treating  the  methods  of  Agreement  and  of  Difference : 
*  Agreement'  he  hardly  distinguishes  from  the  method 
of  simple  enumeration,  which  is  admittedly  problematic ; 
whereas  *  Difference'  he  attempts  to  exhibit  as  strictly 
demonstrative.  In  the  second  place,  he  professes  to 
employ  as  the  'supreme  major  premiss'  for  his  methods 
a  very  wide  but  at  the  same  time  undefined  proposition 
called  the  *Law  of  Causation.'  In  opposition  to  this 
prevalent  view,  I  hold  that  it  is  impossible  to  present 
such  methods  as  those  of  Agreement  and  Difference  as 
strictly  formal  so  long  as  we  attempt  to  subsume  them 
under  so  vague  a  proposition  as  the  Law  of  Causation, 
and  that  each  inference  drawn  in  accordance  with  these 
methods  requires  t^s  own  specific  major  premiss.  The 
formulation  of  such  a  major  premiss  is  the  necessary 
first  step  in  rendering  formally  cogent  any  inference 
(drawn  under  methods  similar  to  Mill's)  from  instances 
finite  in  number,  presented  either  in  passive  observa- 
tion or  under  experimental  conditions.  In  the  third 
place,  whereas  Mill  retains  or  eliminates  a  determining 
factor  according  as  it  affects  or  does  not  affect  a  deter- 
mined character,  in  my  view  the  precise  conclusion  to 
be  drawn  is  not  correctly  expressed  in  terms  of  the 
presence  or  absence  of  factors,  but  rather  in  terms  of 
co-variation,  thus :  according  as  in  two  instances  a  single 
variation  in  any  determining  character  does  or  does  not 
yield  a  variation  in  the  determined  character,  the  same 
will  hold  for  any  and  every  further  variation  of  that 
determining  character. 

§  7.  In  order  to  obtain  the  requisite  premisses  for 
demonstrative  induction,  we  must  assume  that  by  a 
preliminary  inductive  process  based  upon  general  ex- 


f 


DEMONSTRATIVE  INDUCTION 


219 


perience,  a  number  of  variable  circumstances  have  been 
eliminated  as  irrelevant  to  the  formula  to  be  proved. 
The  exposition  of  this  preliminary  process  by  which 
irrelevant  conditions  are  eliminated,  must  be  postponed 
until  we  examine  in  detail  the  nature  of  problematic 
induction.    The  process  itself  must  be  regarded  as  pre- 
scientific;   and  science  takes  up  the  problem  at  the 
point  where  the  character  of  a  phenomenon  is  known  to 
depend  only  upon  a  limited  number  of  variable  con- 
ditions.   This  knowledge  is  expressed  in  a  proposition 
which  constitutes  the  major  premiss  in  the  scientific 
process  which  we  are  about  to  examine  as  a  species  of 
demonstrative   induction.    The   major   in  question  is 
specifically  different  for  different  classes  of  phenomena, 
and  is  in  this  respect  unlike  the  so-called  Law  of  Causa- 
tion which  professes  to  be  the  same  for  every  class  of 
phenomena.    If  the  symbols  A,  B,  C,  D,E  are  taken 
to  illustrate  the  determining  characters,   and  P  the 
thereby  determined  character,  then  the  instances  col- 
lected in  order  to  establish  a  given  generalisation  of 
the  form  ABCDE  ~  P,  must  be  characterised  by  the 
same  set  of  determinables,  and  will  be  said  to  be  of  the 
same  type  or  homogeneous  with  one  another.    The 
specific  major  premiss  may  then  be  expressed  in  the 
formula: 

The  variations  of  the  phenomenal  character  P 
depend  only  upon  variations  in  the  characters  A,  B,C^ 
/?,  E  (say). 

The  conception  of  dependence,  which  the  above 
formula  introduces,  requires  more  precise  explanation. 
In  the  first  place  the  formula  must  be  understood  to 
imply  that  the  variations  of  A,  By  C,  B,  E,  upon  which 


220 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


221 


variations  oiP  depend,  are  independent  of  one  another. 
For  if,  for  example,  a  variation  of  A  entailed  a  varia- 
tion of  B,  then  B  being  a  determined  character  should 
be  omitted  from  amongst  the  determining  characters. 
It  is  only  by  observing  this  principle  that  we  can  apply 
the  essential  rule  for  all  experimentation — that  one  only 
of  the  determining  characters  should  be  varied  at  a 
time.  Again  it  is  essential  that  A,  B,  C,  D,  E,  should 
be  simplex  characters :  for  the  nature  of  the  dependence 
of  P  upon  them  is  such  that,  if  only  one  of  these 
mutually  independent  determining  characters  varies, 
the  character  P  will  vary ;  whereas,  if  more  than  one 
of  them  varied,  P  might  remain  constant.  This  con- 
sideration shows  that  if  any  character  such  as  A  was 
not  simplex,  but  resolvable  into  unknown  factors  X 
and  Y  which  varied  independently  of  one  another,  then 
a  variation  in  A  might  involve  such  a  variation  in 
both  X  and  Y  that  the  character  P  would  remain 
unchanged. 

In  the  second  place,  the  force  of  the  term  'only* 
indicates  that  the  dependence  of  P  upon  A,  B,  C,  Z?,  E 
is  such  that  no  variable  circumstances  other  than  these 
need  be  taken  into  consideration,  all  others  having 
been  previously  eliminated  in  what  we  have  called  the 
prescientific  or  problematic  stage  of  the  induction.  The 
conclusion  that  results  from  this  prescientific  induction 
is  to  be  expressed  by  an  alternation  of  universals  in 
the  form  :  '  Either  every  instance  of  abode  is/,  or  every 
instance  of  abcde  is  p\  or  every  instance  of  abcde  is 
p'\  or.../  From  this  it  follows  that  when  a  single 
instance  is  given  of  abcde  that  is  /,  this  may  be  im- 
mediately universalised  in  the  form  *  Every  abcde  is  /. 


*«i, 


It  should  be  pointed  out  that  this  immediate  univer- 
salisation  is  not  dependent  upon  any  comparison  of  one 
instance  with  another,  and  is  prior  to  the  use  of  such 
methods  as  those  of  difference  or  agreement ;  being  in 
fact  exemplified  above  for  the  case  of  the  atomic  weight 

of  argon. 

The  full  significance  of  the  notion  of  dependence 
is  brought  out  by  taking  not  only  instances  which  agree 
in  the  determining  characters  and  therefore  in  the 
determined  character,  but  by  taking  also  instances 
which  differ  in  the  determining,  and  consequently  also 
in  the  determined  characters.  If  a  variation  in  any  one 
of  the  characters  A,  B,C,  D,  E  entails  a  variation  in 
P,  then,  in  accordance  with  the  principle  underiying 
Mill's  method  of  Difference,  that  character  cannot  be 
eliminated ;  whereas,  if  no  variation  in  P  is  entailed  by 
a  variation  in  some  one  of  the  characters  A.B^Cy  D,  E, 
then,  in  accordance  with  the  principle  underlying  MilFs 
method  of  Agreement,  that  character  can  be  eliminated. 

§  8.  The  forms  of  Demonstrative  Induction  to  be 
now  exhibited  contain  (i)  the  supreme  premiss  of 
dependence  formulated  above  for  a  given  set  of  deter- 
minates, and  (2)  a  finite  set  of  instantial  premisses 
under  the  same  determinates.  These  forms  will  be 
distinguished  under  four  heads  to  be  designated  figures 
rather  than  methods ;  but  will  not  correspond  severally 
to  Mill's  methods,  although  primarily  based  upon  his 
method  of  Difference,  and  with  some  important  modi- 
fications upon  his  method  of  Agreement.  The  notion 
of  ^figure'  is  substituted  for  that  of  'method';  {a)  be- 
cause there  is  only  one  method  employed  in  the  four 
figures,  namely  that  of  varying  one  determining  factor 


f 


222 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


223 


at  a  time ;  and  (d)  because,  as  in  the  case  of  the  figures 
of  syllogism,  the  precise  conclusion  drawn  from  the 
instantial  premisses  will  depend  on  the  nature  of  the 
instances  themselves,  and  the  figure  to  be  employed  in 
any  given  case  will  not  be  foreknown  until  the  instances 
have  been  examined  and  compared.  I  shall  adopt  the 
phrases  'Difference*  and  'Agreement'  for  the  first  two 
figures  but  *  Composition'  and  'Resolution'  for  the  two 
remaining  figures.  All  the  four  figures  have  the  same 
demonstrative  force,  and  the  two  last  figures — though 
they  have  some  resemblance  to  Mill's  or  rather 
Herschel's  method  of  Residues,  which,  as  shown  in  a 
previous  chapter,  is  purely  deductive — have  precisely 
the  same  inductive  nature  as  those  of  Difference  and 
Agreement.  In  each  figure,  the  first  step  in  the  demon- 
strative process  is  to  universalise  each  single  instance 
^aken  separately  in  accordance  with  the  principle  of 
Direct  Universalisation  enunciated  above ;  and  the 
second  to  draw  the  more  specific  conclusion  that  can 
be  inferred  from  a  comparison  of  instances. 

We  proceed  to  give  an  account  of  each  of  the  four 
figures  in  turn. 

§  9.  Figure  of  Difference 

Given  the  supreme  premiss  :  P  depends  only  upon 
ABCDE'.  we  shall  suppose  instantial  premisses  in 
which  variations  occur  in  the  determining  factor  Z?, 
which  is  assumed  to  be  simplex. 

Then  a  single  instance  of  abcde  that  is  p  is  uni- 
versalised  into  'Every  instance  oi  abcde  is  p.* 

Again  a  single  instance  of  abcd'e  that  is  p'  is  uni- 
versalised  into  'Every  instance  oi  abcd'e  is/V 


; 


f 


Comparing  these  two  instances  of  abce,  we  note 
that  a  variation  from  d  to  d'  entails  a  variation  from  p 

top'. 

From  this  we  infer  that  the  value  of  D  is  actually 
operative  in  determining  the  value  of  P.  Hence  any 
further  variation  of  D — say  from  d  to  d" — will  entail  a 
further  variation  of  P — say  from /  to  p".  I.e.  any  value 
of  D  other  than  d  or  d'  will  yield  a  value  of  P  other 

than  /  or  p'. 

Represented  symbolically,  the  conclusion  reached 

is  that 

'Every  instance  of  abcd"e  will  be/"/ 

where  this  universal  is  interpreted  to  signify  that,  within 
the  range  abce,  any  given  difference  in  D  will  entail 
some  difference  in  P,  without  however  indicating  what 
determinate  value  of  P  will  be  yielded  by  the  given 
determinate  value  of  D. 

We  may  symbolise  the  form  of  inference  which  has 
just  been  explained  in  the  following  scheme : 

Figure  of  Difference 
Supreme  Premiss :  P  depends  only  onA.B,  C,  D,  E 
where  D  is  simplex. 

Instantial  Premisses  Immediate  Conclusions 

1.  A  certain  dzi^rrt!'^  is ;>.  .*.  i.  1^^ try  abcde  \s p. 

2.  A  certain  abcd'e  is/'.  .-.  2.  Every  abcd'e  is/. 

Final  Conclusion  :  .'.  Every  abcd"e  is/". 

§  10.  Figure  of  Agreement 

Given  the  supreme  premiss  :  P  depends  only  upon 
ABCDE',  we  shall  suppose  instantial  premisses  in 
which  variations  occur  in  the  determining  factor  A, 
which  is  assumed  to  be  simplex. 


224 


CHAPTER  X 


Then  a  single  instance  of  abcde  that  is  /  is  uni- 
versalised  into  *  Every  instance  oi  abcde  is// 

Again  a  single  instance  of  aJbcde  that  is  /  is  univer- 
salised  into  *  Every  instance  oi  aJbcde  is/.' 

Comparing  these  two  instances  of  bcde,  we  note 
that  a  variation  from  a  to  a'  entails  no  variation  in  P. 

From  this  we  infer  that  the  value  of  A  is  not 
actually  operative  in  determining  the  value  oi  P.  Hence 
any  further  variation  of  A — say  from  a  to  a!^ — will  en- 
tail no  variation  in  P\  i.e.  any  value  of  A  will  yield  the 
same  value  /  of  P. 

Represented  symbolically,  the  conclusion  reached  is 

that: 

*  Every  instance  oi  Abcde  will  yield/,' 

where  this  universal  is  interpreted  to  signify  that  within 
the  range  bcde,  whatever  value  A  may  have,  the  value 
/  will  remain  unaffected. 

We  may  symbolise  the  form  of  inference  which  has 
just  been  explained  in  the  following  scheme : 

Figure  of  Agreement 

Supreme  Premiss:  P  depends  only  on  Ay  By  C,  Z?,  E^ 
where  A  is  simplex. 

Instantial  Premisses  Immediate  Conclusions 

1.  Ps.  ctrtdim.  abcde  \s p.  .'.  I.  Y^^xy  abcdexsp. 

2.  KQ.txX2Sxi  a!bcde\^p.  .',2.  Kwery  a' bcde  is  p. 

Final  Conclusion :  .  • .  Every  d'bcde  is  p, 

§11.  Figure  of  Composition 

Given  the  supreme  premiss  :  P  depends  only  upon 
ABCDE :  we  shall  suppose  instantial  premisses  in 
which  variations  occur  in  the  determining  factor  C, 
which  is  assumed  to  be  simplex. 


DEMONSTRATIVE  INDUCTION 


225 


Then  a  single  instance  of  abcde  that  is  /  is  uni- 
versalised  into  *  Every  instance  oi  abcde  is/.' 

Again  a  single  instance  of  ab/de  that  is/'  is  univer- 
salised  into  *  Every  instance  oi  abc^de  is/V 

Comparing  these  two  instances  of  abde  we  could 
infer,  as  in  the  Figure  of  Difference,  that  a  further 
variation  of  C  would  entail  a  variation  in  /.  But  we 
have  to  contemplate  a  third  instance  where  d'  yields 
the  same  value/  that  was  presented  in  the  first  instance. 
If  the  values  abe  are  known  to  be  the  same  as  in  this 
first  instance,  then  a  difference  in  the  remaining  factor 
d  must  have  accounted  for  the  recurrence  of  the 
same  determined  value  /.  Thus  the  first  and  third 
instances  of  abe  determining  /  must  have  been  due  to 
the  compounding  of  c  with  d  in  the  first  case,  and  to 
the  compounding  of  d^  with  d'^  in  the  third  case.  Such 
a  case  arises  when  the  factor  £>  in  the  third  instance 
has  not  been  amenable  to  precise  evaluation. 

Represented  symbolically  the  conclusion  reached 
is  that : 

*Any  instance  oi  abc'^pe  will  be  d"y 

where  d'^  is  some  unevaluated  value  of  D  other  than 
d  or  d'. 

Symbolically: 

Figure  of  Composition 

Supreme  Premiss:  P  depends  only  onAyB,  C,  Z>,  E, 
where  C  is  simplex. 


Instantial  Premisses 

1.  A  certain  abcde  is;>. 

2.  A  certain  abc'de  isp\ 

Final  Conclusion 


Immediate  Conclusions 
.*.   I.  Kvery  abcde  is  p. 
.'.  2.  Every  abc'de  is p'. 
Every  abcf'pe  is  d". 


J.  L.  II 


15 


226 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


227 


K  12.  Figure  of  Resolution 

Given  the  supreme  premiss  :  P  depends  only  upon 
ABCDE'.  we  shall  suppose  instantial  premisses  in 
which  variations  occur  in  the  determining  factor  E, 
which  is  not  here  assumed  to  be  simplex. 

Then  the  three  single  instances  of 

abcde^p,     abode  <^  p\     abcd^' <^  py 

may  be  respectively  universalised  into 

Every  abcde  is/,  Every  abcde*  is/,  Every  abcde'^  is/. 

Comparing  the  first  and  third  of  these  instances, 
where  under  the  range  abed,  e  and  also  ^"  yield  p,  we 
conclude  that  E  is  complex,  being  resolvable  say  into 
the  two  independent  factors  X,Y\  so  that  (say)  e^xy, 

and^"=yy'. 

Represented  symbolically,  the  conclusion  reached 

is  that 

*  Every  abcdxy  is/,  and  Every  abcdx!';)/'  is  /,' 

where  xy  and  yy  represent  the  resolution  of  e  and  /' 
to  account  for  the  same  value  /  of  P,    Thus  : 

Figure  of  Resolution 
Supreme  Premiss :  /^depends  only  upon  AyB^C,  D,  E. 

Instantial  Premisses  Immediate  Conclusions 

1.  A  certain  abcde  is/.  i.  Every  abcde  is/. 

2.  A  certain  abcd^  is/'.  2.  Every  abcdef  is/'. 

3.  A  certain  abcde''  is/.  3-  Every  abcd^'  is/. 

Final  Conclusion :  E  is  resolvable  into  X  V, 
where  e=xy,  and  e^'^x"/'. 

§  13.    It  will  be  seen  that  each  of  these  figures  of 
inductive  implication  is  formally  equivalent  to  a  single 


disjunction  of  four  propositions.    This  fourfold  disjunc- 
tion may  be  called  : 

T/ie  Antilogism  of  Demonstrative  Induction 

Given  three  instances  of  the  same  type  exhibiting 
three  different  values  of  a  given  determining  character, 
then  no  case  can  arise  in  which : 

(i)  \h^ given  determining  character  is  simplex  ; 

(2)  the  values  of  the  other  determining  characters 
agree  throughout  the  three  instances ; 

(3)  the  value  of  the  determined  character  differs  in 
two  of  the  three  instances  ; 

(4)  the  value  of  the  determined  character  agrees  in 
two  of  the  three  instances. 

Symbolically  expressed,  we  cannot  have 
B  simplex ;  and 


an  instance  oi  a  b  c  d  e  that 
f>  M  y,  a  U  c  d  e  that 
»         „         „   a  U'c  d  e  that 


s  /, 
s/, 
s  /. 


Expressing  this  fourfold  disjunction  in  terms  of  its 
four  equivalent  implications,  we  can  formulate  the 
Figures  of  Demonstrative  Induction  thus: 

(not-4)  Figure  of  Difference  :  If  i  and  2  and  3  ;  then  not-4 

(not-3)  Figure  of  Agreement :  If  i  and  2  and  4 ;  then  not-3 

(not-2)  Figure  of  Composition  :  If  i  and  3  and  4 ;  then  not-2 

(not-i )  Figure  of  Resolution  :  If  2  and  3  and  4 ;  then  not-i. 

In  symbols  this  becomes  : 

Figure  of  Difference. 

\l  B  \s  simplex,  and  we  have 

an  instance  oi  a  b  c  d  e  that  is  /, 
and  an  instance  oi  a  U  c  d  e  that  is  /', 
then  every  instance  of  a  b" c  d  e  will  be  /". 

15—2 


228  CHAPTER  X 

Figure  of  Agreement. 

If  B  is  simplex,  and  we  have 

an  instance  oi  a  b  c  d  e  that  is  /, 

and  an  instance  oi  a  b'  c  d  e  that  is  /, 

then  every  instance  of  a  b"c  d  e  will  be  p. 

Figure  of  Composition, 

If  ^  is  simplex,  and  we  have 

an  instance  oi  a  b  c  d  e  that  is     /, 

and  an  instance  oi  a  b'  c  d  e  that  is     p\ 

then  any  instance  of  a  b"c  p  e  must  be  d'\ 

Figure  of  Resolution. 

If  we  have  an  instance  oi  a  b  c  d  e  that  is  /, 

and  an  instance  oi  a  U  c  d  e  that  is  p', 

and  an  instance  of  a  b" c  d  e  that  is  /, 
then  B  is  complex. 

§  14.  A  simple  illustration  of  the  Figure  of  Dif- 
ference is  afforded  by  Guy-Lussac's  law  which  connects 
variations  in  the  pressure  py  temperature  /,  and  volume 
V,  of  a  specific  gas  g.  Suppose  that  in  two  instances 
without  changing  g  and  p,  a  change  of  temperature 
from  /  to  /'  is  found  to  entail  a  change  of  volume  from 

V  to  1/ .  From  this  it  can  be  inferred  under  the  Figure 
of  Difference  that,  with  the  same  gas  at  the  same 
pressure,  any  further  change  of  temperature,  say  from 
/  to  f\  would  entail  a  further  change  of  volume,  say 

V  to  z/'.  This  experiment  does  not  prove  that  for  any 
other  gas  or  for  any  other  pressure,  a  change  of  tem- 
perature would  entail  a  change  of  volume  ;  nor  does  it 
indicate  what  determinate  value  of  the  volume  would 
be  entailed  by  any  supposed  further  change  of  tempera- 
ture.  It  should  be  observed  that  the  conditions  required 


DEMONSTRATIVE  INDUCTION 


229 


m 


\ 


i 


for  the  Method  of  Difference — namely  precise  con- 
stancy in  all  but  one  of  the  determining  factors— is 
much  more  easily  realisable  when  dealing  with  the 
same  body  or  substance  and  varying  its  alterable  states 
than  when  we  pass  from  one  to  another  body  or  sub- 
stance in  one  of  which  a  character  is  present  and  in 
the  other  absent.  Hence  the  conditions  most  favour- 
able for  the  application  of  the  Figure  of  Difference  are 
those  in  which  concomitant  variations  in  the  determin- 
ing and  determined  factors  are  observed.  For  Mill,  on 
the  other  hand,  the  so-called  Method  of  Concomitant 
Variations  was  primarily  distinguished  from  the  Method 
of  Difference  in  that  the  latter  was  concerned  with 
presence  and  absence,  and  the  former  with  variations 
in  degree.  He  speaks  of  this  method  as  the  one 
necessarily  required  when  we  cannot  wholly  get  rid  of 
a  phenomenon,  and  are  obliged  to  be  satisfied  with 
noting  the  varying  degrees  with  which  it  is  manifested 
from  instance  to  instance  ;  as  if  this  method  were  a 
sort  of  makeshift  which  had  to  be  put  up  with  when 
recourse  to  the  Method  of  Difference  was  impossible. 
But  it  is  precisely  in  those  cases  in  which  we  can  van^ 
the  degree  of  a  phenomenon,  and  not  in  those  that 
can  be  described  as  presence  and  absence,  that  we  can 
be  assured  that  the  rigid  conditions  required  by  the 
Method  of  Difference  are  fulfilled.  Mill  in  adopting 
this  position  neglected  the  consideration  of  the  homo- 
geneity in  any  collection  of  instances  brought  together 
for  comparison  under  any  method  of  induction  what- 
ever. In  the  conception  of  concomitant  variations  is 
included — not  only  quantitative  variations  or  variations 
of  degree  but  also  qualitative  variations  under  any 


230 


CHAPTER  X 


given  determinable  such  as  colour  or  sound.    To  illus- 
trate Concomitant  Variations,  Mill  chose  the  method 
employed  in  connecting  the  varying  heights  of  the 
tides  with  the  variations  of  the  position  of  the  sun  and 
moon   relatively  to  the  earth  ;    but  he  presented  the 
matter  as  if  the  difference  in  the  cogency  of  this  method 
from  that   of   Difference  was  due  to  the  distinction 
between  presence  and  absence  in  the  latter  and  varia- 
tions of  degree  in  the  former ;  whereas  it  is  obvious 
that  the  real  deficiency  in  this  application  of  the  Method 
of  Concomitant  Variations  was  due  to  the  special  nature 
of  the  case,  which  made  it  impossible  to  secure,  in  the 
different  instances  examined,  exact  agreement  in  regard 
to  the  circumstances  not  known  to  be  irrelevant :  e.g. 
the  variations  of  height  of  the  tides  might  have  de- 
pended upon  variations  in  the  force  or  direction  of  the 
wind,  or  in  the  shape  of  the  coast,  etc.    So  far  then 
from  regarding  the  Method  of  Concomitant  Variations 
as  an  inferior  substitute  for  that  of  Difference,  if  by 
the  former  is  meant  variation  in  the  alterable  states  or 
relations  of  some  one  body  or  substance,  and  by  dif- 
ference is  meant  comparison  of  two  similar  bodies  in 
one  of  which  some  quality  is  present  and  in  the  other 
absent,  we  must  regard  the  former  method  as  superior 
to  the  latter.    For  example  :  if  we  attem.pt  empirically 
to  establish  a  causal  connection  between  the  prosperity 
or  the  reverse  of  a  country  and  its  adoption  of  free 
trade  or  protection,  it  would  be  impossible  to  find  two 
different  countries  which  agreed  in  all  relevant  respects 
with    the   exception    of    this    difference   in   industrial 
policy ;  and  hence  a  change  in  which  one  policy  was 
replaced  by  the  other  within  one  and  the  same  country 


DEMONSTRATIVE  INDUCTION 


231 


would  afford  incomparably  more  cogent  evidence  of 
causation  than  a  comparison  of  the  effects  in  two 
different  countries  which  must  necessarily  differ  in 
very  many  respects  that  could  not  be  assumed  to  be 

irrelevant. 

§  1 5.    To  illustrate  the  Figure  of  Agreement  we  may 
take  instances  used  to  establish  the  law  that  the  rate  at 
which  a  body  falls  in  vacuo  to  the  earth  is  independent 
of  its  weight.    In  these  instances  we  keep  unchanged 
all  the  possibly  relevant  circumstances,  such  as  distance 
from  the  earth,  absence  of  air,  substance  and  shape*  of 
the  falling  body,  and  vary  only  the  weight.    From  two 
instances  in  which  the  weight  alone  differs,  we  find 
that  the  time  occupied  in  falling  through  any  given 
distance  is  unchanged.    In  this  way  we  use  the  Figure 
of  Agreement  which  might  also  be  called  the  Figure 
of  Indifference,  since  it  picks  out  a  determining  con- 
dition  which  is  naturally  expected  actually  to  modify 
the  effect  in  question,  and  yet  is  shown  by  a  comparison 
of  instances  to  be  indifferent  as  regards  the  determinate 
value  of  the  effect.    An  illustration  of  this  kind  seems 
not  to  have  occurred  to  Mill,  because  in  his  Method 
of  Agreement  every  circumstance  except  one  differs  in 
the  several  instances ;  whereas,  in  my  formulation  of 
the  corresponding  figure,  every  circumstance  except 
one  agrees  in  the  several  instances.    In  other  words, 
as  regards  the  determining  factors  my  Figures  of  Dif- 
ference and  of  Agreement  require  the  same  condition, 
namely  a  single  difference  ;  whereas  Mill  contrasts  the 
two  by  defining  the  Method  of  Difference  as  involving 
a  single  difference  and  the  Method  of  Agreement  as 
involving  a  single  agreement.    In  fact  Mill  attempts 


232 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


233 


the  elimination  en  bloc  of  all  the  varying  circumstances 
which  distinguish  the  different  instances  in  which  the 
same  effect-value  is  observed,  whereas  what  is  required 
in  order  to  give  corresponding  form  to  the  two  methods 
is  that  we  should  eliminate  as  indifferent  or  irrelevant 
only  one  circumstance  at  a  time. 

§  16.  Having  illustrated  the  figures  of  Agreement 
and  Difference,  I  will  explain  the  strict  procedure 
of  using  these  figures  in  dealing  with  a  number  of 
cause  factors  and  of  effect  factors  conjoined  in  a  set  of 
examined  instances.  Taking  as  our  original  major 
premiss:  ABCDE  ^  PQRT,  i.e.  the  conjunction  of 
the  cause  factors  A,  B,  C,  Z?,  E  determines  the  con- 
junction of  the  effect  factors  />,  Q,  R^  T:  it  is  to  be 
remembered  that  no  cause- factors  other  than  those 
enumerated  are  determinative  of  the  enumerated  effect- 
factors,  as  also  that  no  effect-factors  other  than  those 
enumerated  are  dependent  upon  the  enumerated  cause- 
factors.  We  then  take  in  turn  one  cause-factor  and 
another  and  find  instances  from  which  we  may  conclude, 
in  regard  to  a  given  effect-factor  either,  in  accordance 
with  the  Figure  of  Difference,  that  the  factor  that  is 
varied  is  actually  operative,  or  in  accordance  with  the 
Figure  of  Agreement,  that  such  factor  is  not  actually 
operative :  and  this  procedure  is  repeated  for  each  of 
the  effect-characters  in  turn.  Each  pair  of  instances 
compared  in  this  way  will  lead  to  a  universal  conclusion 
under  the  Figure  of  Difference  or  of  Agreement  as  the 
case  may  be.  This  Complete  method  (as  it  may  be 
called)  is  by  no  means  identical  with  Mill's  Joint 
Method  of  Agreement  and  Difference,  the  use  of  which 
he  advocates  only  to  compensate  for  the  failure  to 


\ 


~* 


secure  variation  in  a  single  factor ;  in  this  Complete 
method,  on  the  other  hand,  one  cause-factor  alone  is 
varied  in  each  pair  of  compared  instances. 

This  process  symbolically  expressed  serves  as  an 
exercise  in  the  application  of  the  principles  underlying 
demonstrative  induction  For  example,  take  the  follow- 
ing instances : 

(i)  abcde  "^ pqrt ;     (2)  a!bcde  <^ pq^t^t ; 

(3)  ab'cde  ^fqr^'t ;     (4)  a'bc'de  ^ffr't ; 

(5)  abcd'e  ^p'q'r'L 

From  (i)  and  (2),  we  eliminate  a  and  ^' as  irrelevant 
to  /  and  t\  and  infer  Abcde  ^ pt.  From  (i)  and  (3), 
we  eliminate  b  and  b'  as  irrelevant  to  q  and  t ;  and 
infer  aBcde  '^  qt.  From  (2)  and  (4),  we  eliminate  c  and 
/  as  irrelevant  to  r  and  / ;  and  infer  a'bCde  '^  r't.  On 
the  other  hand,  from  the  comparison  of  (i)  and  (5)  we 
infer  that  d  and  d^  cannot  be  eliminated  as  ineffective  as 
regards  either  p,  p'  or  q,  (f  or  r,  r' ,  Hence,  under  the 
Figure  of  Difference,  we  infer  abcd"e  ^^  p"q"r^'.  Since, 
however,  in  these  two  instances,  the  variation  of  D  is 
inoperative  on  T,  we  also  infer,  under  the  Figure  of 
Agreement,  abcDe  '^  t.  We  may  now  combine  the 
conclusions  Abcde  ^ pt  and  abcd"e  ~/'yV,  and  thus 
infer  Abccl'^e  ^  p'\  This  conclusion  expresses  the  fact 
that,  under  unchanged  conditions  bee,  A  is  inoperative 
while  D  is  operative  upon  P.  It  should  be  observed 
that  the  conclusion  abcd'^ e  <^ p" q" f^'  is  contrary  to  the 
inference  drawn  by  Mill  in  his  Method  of  Difference  ; 
for,  according  to  his  formulation  of  the  Method,  a 
difference  in  a  single  cause-factor  entails  a  difference 
in    a  single  effect-factor.     Other   inferences — such   as 


234 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


235 


from  (i)  and  (2)  that  a^'bcde<^  q^'r^' — may  be  left  to  the 
ingenuity  of  the  reader  to  discover. 

§  1 7.    Before  illustrating  the  two  remaining  figures, 
it  is  desirable  to  explain  how  the  symbols  employed  in 
my  notation  are  to  be  practically  applied.    When  the 
characters  of  two  or  more  cause-factors  are  represented 
by  such  symbols  as  a.byC,  ...  two  typical  cases  may  arise ; 
( I )  where  a,b,c,.,,  represent  determinates  under  different 
determinables  A,  B,  C...  ;  (2)  where  two  or  more  of 
them  are  determinates  under  the  same  determinable. 
I  n  the  latter  case,  supposing  the  symbol  B  to  represent 
the  same  determinable  character  as  A,  the  three  factors 
a,  b,  c  would  more  naturally  be  symbolised  by  ^,  Uy  c. 
Here  the  recurrence  of  the  symbol  a  indicates  that 
there  are  two  factors  conjoined  which  are  existentially 
different  the  one  from  the  other,  although  characteris- 
able  under  the  same  adjectival  determinable.    In  order 
to  symbolise  the  two-fold  manifestation  of  cause-factors 
characterised  under  the  same  determinable,  we  might 
use  the  subscripts  i,  2,  to  represent  existential  plurality ; 
and  thus,  instead  of  writing  ab,  ab\  alb,  a'b\  etc.,  we 
should  write  a^a^,  a^al,  a{a^y  a^a^y  etc.    For  example, 
the  adjectival  determinable  Force  may  be  represented 
by  Fy  and,  when  two  forces  enter  together  as  cause- 
factors  in  producing  a  certain  effect,  the  possible  varia- 
tions in  which  they  may  be  conjoined  may  be  repre- 
sented  h^  fJ.JJlJlUflfL  etc.    Or  again  :  taking 
the  character  of  a  chemical  element  to  be  indicated  say 
by  its  atomic  weight,  we  may  use  A  to  represent  this 
adjectival  determinable  ;  and  the  possible  variations  in 
which  two  elements  are  conjoined  in  producing  a  com- 
pound may  be  represented  by  a^a^y  a^a^y  a/^o*  ^/^2'>  etc. 


Since,  however,  amongst  symbolists,  any  difference  of 
symbol  such  as  a  and  by  is  never  understood  to  prohibit 
identity  of  meaning,  while  of  course  an  identity  of 
symbol  is  always  understood  to  prohibit  a  difference 
of  meaningy  the  notation  that  I  have  adopted  in  my 
schematisation  of  the  Figures  may  still  be  retained 
without  danger  of  confusion  ;  and,  in  any  case,  it  serves 
to  represent  the  general  principles  of  the  Figures, 
although  in  specific  cases  the  special  notation  indicated 
above  may  be  preferred. 

§  18.  As  regards  the  two  remaining  figures  of 
Composition  and  Resolution,  we  must  point  out  their 
differences  from  the  figures  of  Agreement  and  Dif- 
ference, and  explain  what  is  meant  by  the  composition 
of  causes  as  contrasted  with  the  combination  of  causes. 
Although  these  figures  have  been  exhibited  in  a  form 
according  to  which  their  demonstrative  cogency  is 
equivalent  to  that  of  Difference  or  of  Agreement,  they 
palpably  differ  from  these  latter  in  two  respects.  In 
the  first  place,  the  predicate  of  the  universal  conclusion 
drawn  in  the  last  two  figures  concerns  one  of  the  deter- 
mining factors  D  or  Ey  while  that  in  the  first  two  figures 
concerns  the  determined  factor  P,  In  the  second  place, 
the  last  two  figures  introduce  the  notion  of  *  composi- 
tion '  and  its  converse  *  resolution ' — these  terms  being 
used  in  a  special  and  technical  sense  which  requires 
explanation. 

§  19.  The  notion  of  composition  has  been  long  un- 
derstood in  mathematical  physics,  where  the  resultant  of 
two  directed  forces  regarded  as  components  is  repre- 
sented by  the  diagonal  of  the  parallelogram  whose 
sides  represent  these  components.    The  principle  by 


/ 


j\ 


236 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


237 


which  the  mechanical  effect  of  two  conjoined  forces 
can  thus  be  calculated,  was  contrasted  by  logicians  and 
philosophers  with  the  principle  underlying  chemical 
formulae,  in  which  the  properties  of  a  compound  sub- 
stance could  not  be  calculated  in  terms  of  those  of  the 
elements  combined  in  the  compound.  This  led  to  the 
view  that  there  was  a  fundamental  antithesis  between 
mechanism  and  chemism,  the  former  of  which  involved 
a  *  composition,'  the  latter  a  '  combination '  of  cause- 
factors.  Mill  introduced  and  explained  the  phrase 
*  composition  of  causes  '  (i.e.  of  cause-factors)  and  con- 
trasted this  with  the  combination  of  cause-factors, 
specially  characteristic  of  chemical  phenomena,  and 
also,  in  his  opinion,  of  many  psychological  and  socio- 
logical phenomena.  Mill's  explanation  is  not  altogether 
satisfactory.  I  will  therefore  attempt  my  own  ex- 
planation of  the  antithesis  between  composition  and 
combination. 

When  two  cause-factors  represented,  say,  by  the 
determinables  B  and  C  are  such  that  there  are  certain 
pairs  of  values,  say  be  and  Ud,  which  jointly  determine 
the  same  value/  of  an  effect  character  P\  then,  referen- 
tially  to  P,  the  conjunction  reconstitutes  a  composition. 
On  the  other  hand,  when  there  are  no  pairs  of  values 
under  the  determinables  B  and  C,  such  as  be  and  ^V, 
which  jointly  determine  the  same  value  of  P\  then, 
referentially  to  P,  the  conjunction  BC  constitutes  a 
combination.  What  is  important  to  note  here  is  that 
the  distinction  between  composition  and  combination 
is  not  absolute ;  for  certain  conjunctions  of  cause- 
factors  may  constitute  a  composition  referentially  to 
one  assigned   effect-character,  and  a  combination   re- 


ferentially to  another.  For  example,  when  chemical 
elements  are  conjoined  in  producing  a  compound  sub- 
stance, it  is  possible  to  take  the  weights  of  certain 
elements  and  different  weights  of  other  elements  so  as 
to  produce  a  compound  of  the  sam.e  weight ;  hence 
referentially  to  the  effect  weight,  the  conjunction  of 
chemical  elements  comes  under  the  principle  of  com- 
position. But  as  regards  the  chemical  character  of  the 
elements  conjoined,  it  is  impossible  so  to  vary  these  as 
to  produce  a  compound  of  the  same  chemical  character 
in  two  different  cases ;  for  instance,  the  substance 
having  the  chemical  properties  of  water  can  only  be 
produced  by  the  combination  of  hydrogen  and  oxygen. 

This  account  of  the  distinction  between  composition 
and  combination  is  to  be  regarded  as  an  indication 
rather  than  as  a  definition.  Expressed  mathematically: 
the  conjunction  of  the  factors  B  and  C  constitute  a 
composition,  referentially  to  the  effect  P,  when  there 
is  a  certain  function /"such  that/  equals /(^,  c)  for  any 
and  every  value  b  and  c  oi  B  and  C.  We  might  there- 
fore replace  the  terms  composition  and  combination 
respectively  by  the  more  suggestive  terms  functional 
and  non-functional  conjunction.  The  method  of  dis- 
covering and  establishing  such  functional  relations  will 
be  treated  in  the  next  chapter.  But  we  cannot  well 
illustrate  the  figures  of  Composition  and  Resolution 
without  first  modifying  their  formulation  in  view  of  the 
above  explanation  of  the  nature  of  composition. 

§  20.  In  the  figure  of  Composition  as  symbolically 
formulated,  we  took  two  instances  agreeing  as  regards 
the  determining  factors  abde,  and  a  third  instance 
agreeing  with  both  as  regards  abe,  but  in  which  the 


238 


CHAPTER  X 


DEMONSTRATIVE  INDUCTION 


239 


factor  D  was  unamenable  to  precise  calculation.    We 
then  supposed  that,  while  in  the  first  two  instances  the 
differences  c  and  d  in  the  determining  factor  C  yielded 
a  corresponding  difference  /  and  /'  in  the  determined 
factor  P\  yet,  in  a  third  instance,  c^^  yielded  the  unex- 
pected effect  p  equivalent  to  that  yielded  in  the  first. 
The  unexpectedness  of  this  result  was  thus  accounted 
for  either  by  our  inability,   in   the   third  instance,  to 
measure  the  factor  Z>,  or  by  our  error  in  supposing  that 
its  value  was  still  unchanged.    Now,  instead  of  illus- 
trating our  figure  by  supposing  equivalence  as  regards 
P  in  the  first  and  third  instance — a  somewhat  artificial 
assumption — let  us  suppose  rather  that  in  the  third 
instance  the  effect,  say/3,  was  other  than  that  calculated 
by  a  foreknown  formula  in  which  the  value  of  P  would 
be  given  by/''=/(^,  b,  d' ,  e).    On  the  assumption  that 
the   correctness   of   this   formula   had   been   properly 
assured  by  means  of  the  functional  extension  of  the 
Figure   of   Difference  or  of  Concomitant  Variations, 
we  should  rightly  infer  that  any  instance  of  abd'p^e 
would  entail  d"  in  place  of  d,  so  that  the  effect /g,  under 
the  constant  conditions  abe,  would  be  due  to  the  com- 
position (!'d'\  and  not  merely  to  d^ 

In  this  modified  form,  the  Figure  of  Composition 
can  be  illustrated  by  the  irregular  motions  from  /  to  /, 
of  the  planet  Uranus,  the  positions  a,  b,  e,  of  any  other 
planets  being  effectively  unaltered  while  that  of  the  sun 
had  changed  from  c  to  d'.  The  motion  from  d  to  d'^  of 
an  unknown  planet,  afterwards  called  Neptune,  con- 
joined with  that  of  the  sun  from  c  to  d^  accounted  for 
the  unexpected  movement  of  Uranus  from  p  to  p^]  in 
other  words,  a,  b,  e  being  constant,  p^  was  the  same 


< 


function  of  d^  and  d'^  as  p  was  of  c  and  d\  so  that  P 
was  a  function,  not  of  C  alone,  but  of  C  and  D  com- 
pounded. 

A  similar  illustration  of  the  Figure  of  Resolution 
is  found  in  the  experiments  by  which  the  newchtimical 
substance  argon  was  discovered  by  Sir  William  Ramsay. 
Here  the  factor  E  would  represent  atmospheric  nitro- 
gen, and  its  greater  weight — as  conijicired  with  that 
of  nitrogen  prepared  from  chemical  comiK)unds--was 
accounted  ior  by  the  resolution  of  the  atmospheric 
nitrogen  into  the  two  components  arL^oii  and  pure 
nitrogen.  It  should  be  pointed  out  that  the  resolution 
here  employed  was  not  a  chemical  analysis,  for  argon 
does  not  combine  with  any  other  element  (as  far  as  is 
at  present  known)  and  therefore  the  rt  -ohition  in 
question  was  a  true  instance  of  the  converse  of  com- 
position. 

In  regard  to  the  illustration  o{ Composition  involving 
the  discovery  of  Neptune,  and  that  of  Resolution  in- 
volving the  discovery  of  argon,  the  precise  measure- 
ments finally  made  reduced  the  inference  to  a  purelv 
deductive  form,  which  assumed  the  character  of  tht! 
method  of  Residues  according  to  my  interpretation  of 
this  method  (see  p.  118). 


I 


FUNCTIONAL  INDUCTION 


241 


CHAPTER  XI 

THE  FUNCTIONAL  EXTENSION  OF  DEMONSTRATIVE 

INDUCTION 

§  I.  In  concluding  the  treatment  of  demonstrative 
inference  I  propose  to  recapitulate  the  results  that  have 
been  so  far  reached,  and  to  bring  into  focus  the  dis- 
tinctions and  connections  between  the  several  forms  of 
inference,  deductive,  inductive  and  problematic.  I  have 
already  examined  the  general  notion  of  function,  and 
shown  how  it  is  employed  in  mathematical  and  other 
processes  of  deductive  inference ;  and  it  remains  to 
exhibit  this  notion  as  it  enters  into  inductive  inference — 
this  constituting  the  specifically  new  topic  to  be  dis- 
cussed in  the  present  chapter. 

Pure  induction,  by  which  is  to  be  understood  that 
which  involves  no  assumption  of  universal  laws,  has 
been  shown  to  be  the  sole  direct  and  ultimate  mode  of 
generalising  from  instances  examined  and  theoretically 
enumerable.  This  species  of  induction  I  have  called 
problematic  because,  in  my  view,  the  universal  proposi- 
tions which  it  establishes  must  be  regarded,  not  as 
absolutely  certified,  but  as  accepted  only  with  a  higher 
or  lower  degree  of  probability  depending  upon  the 
collective  character  of  the  instances  enumerated.  The 
possibility  of  establishing  such  direct  generalisations 
depends  upon  certain  postulates,  the  discussion  of  which 
raises  one  of  the  most  important  and  difficult  problems 
of  philosophical  logic  ;  and  even  then,  the  probability 


ii 


to  be  attached  to  generalisations  thus  established  has 
to  be  determined  by  reference  to  the  formal  principles 
of  probability.  But,  so  far  as  these  generalisations 
enter  into  the  account  of  demonstration,  they  function 
as  major  premisses.  Demonstrative  induction,  then,  so 
far  rt-senibles  deduction  in  tha.!  it  requires  the  coniunc- 
tion  of  two  types  of  premisses:  (i)  the  major  or  supreme 
universal  premiss,  which  expresses  the  relation  of 
dependence  between  one  specified  set  of  variables 
and  another;  and  (2)  the  minor  or  instantial  |irtmiss 
which  sums  up  the  results  of  single  observations  or  ex- 
periments. The  major  premiss  in  this  mixed  form  of 
demonstration  is  formulated,  not  as  a  uniformity  per- 
vading all  nature,  but  as  a  specified  universal  holding 
only  for  the  special  class  of  phenomena  to  which  the 
conclusion  refers. 

§  2.  A  very  general  statement  of  the  contrast 
between  my  exposition  and  Mill's  is  conveniently  intro- 
duced at  this  point.  I  have  deliberately  separated  the 
treatment  of  formal  or  demonstrative  induction  from 
that  of  problematic  induction.  In  the  latt  r.  the  a  cu- 
mulation of  instances  is  all  important ;  in  the  fVinner, 
a  precise  major  premiss,  relating  to  a  finite  and  (nu- 
merable set  of  determinates,  is  required  in  f  ach  tep 
of  the  formal  process.  These  major  premisses  are 
assumed  to  have  been  previously  establislied,  with  a 
higher  or  lower  degree  of  probability,  on  the  principles 
of  problematic  induction.  The  essence  of  iirobkmatic 
as  contrasted  with  formal  induction  is  expressed  in 
three  statements:  first;  no  wide  generalisation,  such 
as  that  which  asserts  the  uniformity  of  nature,  is  in- 
volved ;  secondly ;  the  instances  compared  are  not 
J.  L.  n  16 


I 


I 


242 


CHAPTER  XI 


FUNCTIONAL  INDUCTION 


243 


determinately  analysed  with  respect  to  the  variable 
characters  upon  which  the  proposed  generalisation 
may  depend  ;  and  hence,  thirdly,  an  indefinite  multi- 
plication of  instances  is  required  in  order  to  give  any 
appreciable  value  to  the  probability  of  the  conclusion. 
It  is  partly  for  this  reason  that  Mill's  account  of  the 
Method  of  Agreement  differs  so  considerably  from  my 
extremely  simple  Figure  of  Agreement ;  for  Mill  is 
largely  thinking,  under  the  title  Agreement,  of  a  direct 
method  of  establishing  empirical  generalisations  to 
which  only  an  inferior  degree  of  probability  can  be 
attached.  The  generalisations  thus  established  by 
problematic  induction  function  as  major  premisses  in 
demonstrative  processes  in  one  of  two  ways  :  either  as 
established  with  what  may  be  called  experiential  as 
opposed  to  rational  certitude  ;  or  as  put  forward  hypo- 
thetically,  and  thus  as  exhibiting  forms  of  implication 
rather  than  of  inference — implication  being  defined, 
as  in  Chapter  I,  to  be  potential  or  hypothetical 
inference. 

§  3.  The  term  hypothesis  has  been  used  by  logicians 
in  so  very  many  senses  that,  in  order  to  obviate  logical 
confusion,  it  will  be  well  to  examine  its  various  usages, 
showing  how  they  have  developed  from  one  funda- 
mental element.  This  element  will  be  found  to  be 
definitely  epistemic  rather  than  constitutive,  and  for 
my  own  purposes  I  consequently  prefer  to  use  the 
phrase  '  hypothetically  entertained/  which  has  an  epi- 
stemic significance  quite  independent  of  the  form  or 
content  of  the  proposition  so  entertained.  We  may 
take  in  turn  the  various  meanings  of  the  substantive 
*  hypothesis  *  or  the  adjective  '  hypothetical '  that  occur 


r 


V 


in  deductive  or  inductive  logic,  in  order  partly  to 
connect  and  partly  to 'contrast  its  epistemic  with  its 
other  bearings.  In  traditional  formal  logic,  propositions 
are  called  hypothetical  which  are  in  fact  compounded 
out  of  two  categorical  propositions,  say/  and  q,  in 
this  case,  while  the  adjective  hypothetical  is  traditionally 
used  to  denote  a  particular  species  of  compound  pro- 
position, namely  that  of  the  form  ' if  p  then  q'\  yet  at 
the  same  time  the  term  hypothesis  clings  firstly  to  the 
proposition  p  because  in  this  form  it  is  not  actually 
asserted,  and  next  to  the  proposition  q  because  it  is 
only  assertible  on  condition  that  p  has  been  asserted. 
Thus  the  adjective  hypothetical  is  actually  attached  to 
three  quite  distinct  propositions  or  forms  of  proposition : 
.the  compound  '  if  /  then  q ' ;  the  simple  proposition  / 
itself,  which  I  call  the  implicans ;  and  the  simple  pro- 
ppsition  q  which  I  call  the  implicate.  Now  in  order  to 
make  a  first  approximation  to  justifying  this  confused 
terminology,  we  must  consider  its  epistemic  aspect,  and 
we  may  say  that  normally  both  the  implicans  separately 
and  the  implicate  separately  are  entertained  hypo- 
thetically, while  the  compound  proposition  *  \{ p  then  q ' 
is  entertained  assertorically.  Hence,  even  where  the 
term  hypothetical  is  used  in  its  most  precise  technical 
sense,  it  is  applied  to  a  form  of  proposition  assumed 
to  be  entertained  assertorically,  the  components  alone 
of  this  assertoric  compound  being  entertained  hypo- 
thetically. 

The  recognition  of  this  ambiguity  in  the  use  of  the 
term  hypothetical  resolves  the  often  disputed  problem 
of  the  relation  in  general  between  induction  and  deduc- 
tion.   When  we  are  concerned  with  the  purely  formal 

16 — 2 


\^i 


% 


/ 


I 


244 


CHAPTER  XI 


FUNCTIONAL  INDUCTION 


relation  of  implication  as  subsisting  between  the  pre- 
misses and  conclusion  of  any  argument  of  the  general 
nature  of  a  syllogism,  then  these  premisses  need  only 
be  entertained  hypothetically;  while,  at  the  same  time, 
the  relation  of  implication  itself  is  to  be  conceived,  not 
only  as  assertorically  advanced,  but  even  as  having  the 
highest  degree  or  kind  of  assertoric  certitude.  The 
conclusion  of  a  syllogism  thus  deduced  is  usually  spoken 
of  as  demonstrated,  i.e.  as  having  demonstrative  certi- 
tude ;  although,  taken  by  itself,  any  kind  or  degree  of 
certainty  attaching  to  it  is  wholly  dependent  upon  the 
kind  or  degree  of  certitude  with  which  the  premisses 
are  entertained.  Taking  full  advantage,  then,  of  Mill's 
account  of  the  functions  and  value  of  the  syllogism, 
we  may  say  that  the  hypothetical  conclusion  has  been 
hypothetically  demonstrated,  and  can  only  be  asse^r- 
torically  demonstrated  when  we  have  examined  aiid 
tested  the  truth  of  the  premisses.  Only  when  the  major 
premiss  has  been  inductively  established  can  the  con- 
clusion be  entertained  categorically,  and  even  then 
with  a  degree  of  probability  dependent  upon  that  of 
the  major  premiss ;  and  ultimately  upon  the  mode  of 
induction  by  which  the  major  has  been  established. 

§  4.  The  problematic  nature  of  the  universal  ob- 
tained by  induction  and  functioning  as  major  premiss 
in  a  deductive  process  has  led  to  a  confusion  between 
the  notions  problematic  and  hypotheticaly  resulting  in 
the  use  of  the  term  'hypothesis'  for  any  proposition 
entertained  with  a  degree  of  probability.  Thus,  when 
Jevons  says  that  all  induction  is  hypothetical,  what  he 
means  is  merely  that  an  inductive  conclusion  has  not 
certainty  but  probability.    Thus  any  inductive  generali- 


245 


'"*&, 


Q 


*■*. 


1*^ 


sation  is  commonly  called  a  hypothesis ;  and  the  term 
1  when  applied  to  a  scientific  theory  may  have  three 
alternative  meanings  :  first,  it  may  mean  that  the  pro- 
position is  unproven  ;  secondly,  that  the  proposition 
has  an  appreciable  degree  of  probability  which  renders 
it  worth  considering ;  thirdly,  that  the  proposition  has 
no  appreciable  probability  at  all,  and  may  even  be 
known  to  be  false.  Besides  the  epistemic  significance 
revealed  in  all  these  three  alternative  meanings,  the 
\  term  hypothesis  must  also  be  understood  to  indicate 
j  the  purpose  which  an  unproven  universal,  definitely 
formulated,  fulfils  in  calculating  deductively  the  con- 
clusions to  which  it  would  lead.  In  fact  Jevons,  in 
V  describing  induction  as  hypothetical,  uses  the  term  in 
two  quite  different  senses  :  first,  in  the  formal  sense,  to 
inldicate  the  provisional  or  tentative  attitude  towards  a 
ui^iversal  before  we  have  confirmed  it  by  a  process  in- 
vcplving  deduction ;  and,  secondly,  to  represent  the 
firlial  attitude  towards  a  universal  after  it  has  been 
te'Sted  and  confirmed  with  the  highest  attainable  degree 
of  probability.  With  the  view  indicated  in  the  second 
.  application  of  the  term  hypothesis,  I  agree ;  but,  as 
regards  the  first  use  of  the  term,  it  seems  to  me  that 
we  always  adopt  a  tentative  attitude  towards  a  proposi- 
tion entertained  as  a  proposal,  whether  it  is  to  be  proved 
deductively  or  inductively;  so  that  the  term  as  applied 
to  a  proposition  to  be  proved  does  not  represent  any 
characteristic  peculiar  to  induction.  Now  the  special 
topic  with  which  this  chapter  is  concerned  involves 
both  the  contrasted  ideas  of  hypothesis :  namely,  of  a 
proposition  having  a  certain  degree  of  probability,  and 
of  one  put  forward  to  be  tested  by  appropriate  evi- 


246 


CHAPTER  XI 


FUNCTIONAL  INDUCTION 


dence.  Thus,  while  the  functional  formula  in  deduction 
is  assumed  to  be  true  and  therefore  may  serve  as  pre- 
miss for  deducing  an  equally  assured  conclusion,  the 
inductive  aspect  of  such  a  functional  formula  presents 
the  inverse  problem  ;  for  we  have  now  to  examine  by 
what  kind  of  instances,  and  by  what  modes  of  com- 
parison, the  functional  formula  itself  can  be  established. 
So  far  as  this  process  of  examination  may  be  said 
to  have  a  special  characteristic  by  which  it  may  be 
distinguished  from  problematic  induction  used  for 
establishing  the  wide  generalisations  of  science,  its 
peculiarity  is  that  a  comparatively  small  number  of 
instances  will  constitute  the  sufficient  factual  basis  for 
the  establishment  of  the  formula,  and  that  the  actual 
procedure  of  mathematical  physics,  at  least  in  thp 
majority  of  cases,  rightly  attaches  practical  certitudje 
to  the  formula  thus  inferred.  j 

§  5.  In  order  to  show  how  the  functional  formula 
is  established,  I  must  refer  to  my  account  of  the  figures 
of  Demonstrative  Induction.  There  the  conclusion 
demonstratively  drawn  does  not  assign  the  specific 
value  of  the  effect-character  that  is  to  be  correlated 
with  any  given  value  of  a  cause-character.  In  popular 
language,  the  conclusions  drawn  would  be  termed 
qualitative  not  quantitative  ;  that  is  to  say,  the  figures 
establish  causal  connection  without  determination  of  a 
causal  law  or  formula.  I  n  comparing  the  different  figures, 
it  is  seen  that  the  Figure  of  Difference,  which  stands 
first,  is  a  direct  expression  of  the  principle  of  the  de- 
pendence of  change  in  the  effect  upon  change  in  the 
cause;  and  that  the  Figure  of  Agreement  or  of  Indif- 
ference is  complementary  to  that  of  Difference  in  the 


247 


■4 


/ 


I 


i 

1, 


same  sense  as  the  universal  or  implicative  *if  not-/  then 
not-^ '  is  the  complementary  of  *  if  /  then  q ' ;  while 
the  Figures  of  Composition  and  Resolution  merely 
carry  out  the  principle  of  Difference  under  certain 
more  complicated  circumstances.  There  is,  therefore, 
one  principle  common  to  all  the  four  figures,  namely 
that  underlying  the  Figure  of  Difference — the  functional 
extension  of  which  will  be  our  principal  concern. 

The  original  formula  of  Difference  may  be  restated 
in  the  following  canon  :  When  in  two  instances  a  dif- 
ference in  the  cause-character  D  entails  a  difference  in 
the  effect-character  P,  all  other  cause-characters  which 
might  contribute  to  the  determination  of  P  being  the 
same  in  the  two  instances,  then  we  infer  that  any  other 
difference  in  the  cause-character  will  be  correlated  with 
some  other  difference  in  the  effect-character,  under  the 
continued  constancy  of  the  remaining  cause-characters. 
Now  this  canon,  which  applies  to  two  instances  only, 
may  be  obviously  extended  to  any  number  of  instances 
all  of  which  conform  to  the  figure  of  Difference :  i.e. 
all  other  cause-factors  remaining  unchanged,  we  find  a 
series  of  instances  in  which  D  alone  varies,  and  in 
which  the  determinate  values  d,  d\  d'\  d"\  etc.,  say, 
are  associated  respectively  with  /,  p\  p'\  p"\  etc.  Now, 
as  in  the  simple  case  of  two  instances,  these  observa- 
tions do  not  enable  us  to  assign  the  specific  value  of  P 
that  is  to  be  correlated  with  any  given  value  of  D : 
we  can  still  only  infer  that  any  further  change  in  D 
will  be  associated  with  some  further  change  in  P,  The 
required  extension  of  the  figure  of  Difference  consists, 
therefore,  in  the  determination  of  P  as  a  function  of  I  J 
which  shall  hold  for  all  unexamined  as  well  as  examined 


\i 


248 


CHAPTER  XI 


FUNCTIONAL  INDUCTION 


249 


instances.  A  famous  example  of  the  determination  of 
such  a  function  is  that  formulated  by  Kepler  who,  after 
nineteen  guesses,  discovered  a  formula  for  the  plane- 
tary movements  about  the  sun  which  co-ordinated 
the  spatio-temporal  relations  for  the  cases — necessarily 
finite  in  number — that  he  was  able  to  examine  and 
measure.  The  discovery  of  this  formula  involved  nothing 
of  the  nature  of  inductive  inference,  but  its  application 
to  all  the  planetary  positions  intervening  between  those 
observed  constituted  a  genuine  inductive  inference,  so 
easy  to  draw  that  neither  Whewell  nor  Mill  seems  to 
have  been  aware  that  any  such  inference  was  implicitly 
made. 

The  canon  for  the  Figure  of  Composition  may  be 
reformulated  as  follows :  When  in  several  instances 
variations  in  the  single  cause-character  C  have  entailed 
variations  in  the  effect-character  P  such  that,  in  ac- 
cordance with  the  functional  extension  of  the  Figure 
of  Difference,  P  has  been  shown  to  be  a  certain  func- 
tion of  C,  then,  if  some  similar  instance  of  a  further 
variation  of  C  has  entailed  a  variation  oi  P  not  satisfying 
this  function,  we  infer  that,  in  this  instance,  besides 
C  some  other  character,  say  D,  has  varied,  and  hence 
that  P  depends  upon  the  composition  of  C  with  D, 
This  simple  use  of  the  Figure  of  Composition  does 
not,  however,  enable  us  to  determine  the  value  of  D  in 
the  particular  instance  observed.  In  expanding  this 
figure  therefore  we  have  to  look  for  further  instances 
in  which  both  C  and  D  can  be  evaluated ;  and  thus 
construct  a  formula  by  which  P  is  represented  as  a 
function  both  of  C  and  of  D.  This  method  should  be 
compared  with  that  of  Residues,  which  I  have  regarded 


% 


d 


V 


\ 


^' 


w 

Ik 


as  purely  deductive ;  for,  in  the  method  of  Residues, 
the  values  of  D  are  determined  deductively  from  the 
known  formula  p  -fie,  d\  whereas,  in  our  extension  of 
the  Figure  of  Composition,  the  formula  p—f{c,  d)  is 
determined  inductively  from  the  observed  values  of  D, 
The  case  of  the  irregularities  in  the  movements  of 
Uranus,  instanced  in  the  previous  chapter,  illustrates 
this  type  of  functional  extension. 

§  6.  Now  the  formula  which  expresses  an  effect  as 
a  function  of  one  or  more  cause-factors  must  at  least 
satisfy  the  negative  condition  that  it  fits  all  the  examined 
instances  as  regards  the  observed  values  of  cause  and 
effect.  Many  logicians,  and  certainly  many  experi- 
menters in  practical  branches  of  science,  are  finally 
satisfied  with  this  negative  criterion.  They  assert,  in 
effect,  that  provided  the  formula  p=f{d),  where /has 
some  specific  form,  agrees  with  the  values  of  P  and  D 
as  measured  in  the  examined  cases,  then  it  has  all  the 
guarantee  that  experimentation  requires  for  its  uni- 
versalisation.  But  the  mathematician  points  out  that, 
theoretically  speaking,  there  are  an  infinity  of  different 
functions  that  would  exactly  fit  any  finite  number  of 
cases  of  covariation.  Hence  he  demands  in  general  a 
much  more  rigid  defence  for  selecting  one  formula 
rather  than  another  to  represent  the  universal  law. 

In  order  to  escape  this  threatening  annihilation  of 
inductive  inference,  we  may  indicate  two  fundamental 
principles  upon  which  the  highest  attainable  degree  of 
certainty,  which  may  be  called  practical  or  experiential 
certitude,  depends.  In  the  first  place,  reliance  is  placed 
upon  the  character  of  the  formula  itself,  and  in  par- 
ticular on  its  comparative  simplicity  ;    in  the  second 

16-5 


^•1 


250 


CHAPTER  XI 


FUNCTIONAL  INDUCTION 


251 


place,  the  higher  credibility  of  a  proposed  formula 
depends  upon  its  analogies  with  other  sufficiently  well- 
established  formulae  in  similar  classes  of  phenomena. 
Briefly,  the  criteria  of  simplicity  arid  analogy,  especially 
when  conjoined,  confer  upon  a  formula  of  covariation 
that  highest  degree  of  probability  which  allows  us  to 
regard  the  induction,  not  as  merely  problematic,  but  as 
virtually  demonstrative.  For  example,  the  experiments 
that  have  been  conducted  in  regard  to  the  covariations 
of  temperature,  pressure  and  volume  of  gases  have 
always  been  treated  by  physicists  as  conferring  absolute 
demonstrative  certitude  upon  the  formulae  inferred, 
although  they  have  been  actually  confirmed  from  a 
necessarily  limited  number  of  observations. 

We  may  illustrate  the  notion  of  simplicity  by  taking 
the  simplest  of  all  possible  functions,  namely  where  p 

is  proportional  to  d,  or  its  inverse  -%.    For  example,  if 

we  have  instances  in  which,  weight  being  the  deter- 
mined factor,  and  some  quantitatively  measurable  cause 
D  varies  so  that  where  we  double  D  we  double  P,  and 
where  we  treble  D  we  treble  P,  and  so  on  for  fractional 
as  well  as  integral  multipliers,  we  inductively  infer  that 
P,  not  merely  varies  with  D,  but  in  mathematical 
language,  varies  as  D.  There  have  been  philosophers 
who,  in  effect,  have  imagined  that,  unless  a  causal 
formula  can  be  expressed  by  a  proportionate  relation 
of  cause  to  effect,  it  must  be  regarded  as  a  mere  em- 
pirical rule ;  and  conversely,  as  soon  as  instances  are 
found  to  fit  some  such  simple  formula,  the  generalisa- 
tion may  be  regarded  as  absolutely  certified.  A  slightly 
less  simple  kind  of  formula  is  exemplified  by  gravitation 


f 


41 


:%: 


where,  for  a  given  attracting  mass,  the  acceleration  of 
the  attracted  body  varies  inversely  as  the  square  of 
the  distance,  being  in  the  direction  towards  the  attract- 
ing body.  The  high  probability  of  this  formula  is  due, 
not  only  to  its  relative  simplicity,  but  to  its  analogy 
with  the  independently  known  formula  for  the  intensity 
of  radiant  light  or  heat.  Moreover  the  formula  in 
question  could  have  been  deduced  from  the  assumption 
that  radiation  operates  equally  in  all  spatial  directions, 
so  that  its  magnitude  upon  any  part  of  a  spherical  sur- 
face is  inversely  proportional  to  the  area  of  that  surface 
and  therefore  to  the  square  of  the  distance.  In  the 
examples  thus  brought  forward,  indications  are  given 
of  the  kind  of  reasoning  upon  which  the  high  proba- 
bility attached  to  any  formula  that  fits  the  examined 
instances  is  based. 

§  7.  The  criterion  of  simplicity  is  not  often  directly 
applicable ;  but,  when  in  a  relatively  complex  conjunc- 
tion of  circumstances  that  can  be  analysed,  a  formula 
is  constructed  that  could  have  been  deduced  from  a 
combination  of  wider  and  well-established  formulae  of 
comparative  simplicity,  then  an  empirical  formula  thus 
confirmed  acquires  problematic  value  corresponding  to 
that  of  the  laws  from  the  combination  of  which  it  could 
have  been  deduced.  Both  Whewell  and  Mill  have 
taken  this  kind  of  criterion  as  fundamental  in  their 
theories  of  induction  ;  Whewell  using  the  phrase  *  con- 
silience of  inductions,'  and  Mill  having  in  his  earlier 
chapters  put  forward  this  deductive  confirmation  as  the 
one  principle  dominating  his  whole  theory.  At  first 
sight  Mill's  position  is  paradoxical,  since  he  apparently 
attributes  a  higher  probability-value  to  a  law,  merely 


252 


CHAPTER  XI 


FUNCTIONAL  INDUCTION 


253 


on  the  ground  of  its  width,  whereas  it  would  appear 
that  the  narrower  generalisation  is  the  safer.  I  think, 
on  this  matter,  we  must  recognise  the  value  of  the  two 
opposed  principles  that  have  been  put  forward.  On 
the  one  hand,  mere  simplicity  has  been  elevated  into 
a  supreme  criterion  ;  but,  so  far  from  admitting  that 
simplicity  alone  guarantees  a  formula,  we  must  main- 
tain that  where  a  known  complexity  of  circumstances  is 
involved,  a  corresponding  complexity  must  be  expected 
to  characterise  their  co-ordinating  formula.  Hence, 
when  a  class  of  phenomena  that  have  not  been  defini- 
tively analysed  resembles  other  classes  for  which  a 
complex  formula  has  been  established,  a  corresponding 
complexity  should  be  anticipated  for  the  given  class  ; 
whereas  the  formula  for  a  class  of  phenomena  analogous 
to  others  for  which  a  simple  formula  holds  may  rightly 
be  expected  to  be  simple.  The  criterion  of  simplicity, 
when  including  its  indirect  as  well  as  its  direct  form,  is 
of  value  ;  but  it  is  only  when  analogy  is  thus  conjoined 
with  simplicity  that  we  may  attach  practical  certitude 
to  a  formula  which  satisfies  at  least  the  negative  cri- 
terion of  fitting  perhaps  only  a  small  number  of  well- 
examined  cases. 

§  8.  The  theory  of  what  I  have  called  the  functional 
extension  of  demonstrative  induction  constitutes  a  link 
between  the  Demonstrative  and  the  Problematic  forms 
of  inference.  For  certain  rules  (of  a  strictly  formal 
character)  are  required  for  deducing,  amongst  all  the 
functions  which  fit  the  observed  co-variations,  the  most 
probable  function  of  the  variable  cause-factors  by  which 
an  effect-factor  may  be  calculated.  The  oldest  and  most 
usual  method  of  determining  this  function  is  known  as 


I 

4 


f 


I 


the  method  of  least  squares.  Its  validity  depends  upon 
a  certain  assumption  with  regard  to  the  form  of  the 
Law  of  Error,  i.e.  of  the  function  exhibited  bv  diver- 
gences  from  a  mean  or  average,  when  the  number  of 
co-variational  instances  is  indefinitely  increased  ;  and  a 
different  method  must  be  employed  for  each  correspond- 
ing different  assumption.  The  reader  must  be  referred 
to  Mr  J.  M.  Keynes's  Treatise  on  Probability,  Chapter 
XVII,  for  a  very  comprehensive  and  original  discussion 
of  this  topic. 

The  inductive  inference  examined  in  the  above  is 
thus  shown  to  be  based  upon  purely  formal  and  demon- 
strative principles  of  probability,  whereas  the  discussion 
of  problematic  induction  to  be  developed  in  Part  III 
will  introduce  informal  theorems  of  probability,  Ij  i>t  d 
on  postulates  of  a  highly  controversial  nature,  it  is 
therefore  legitimate,  and  even  necessary,  to  include  the 
functional  extension  of  the  figures  of  induction  under 
the  general  title  of  demonstrative  inference. 


INDEX 


255 


INDEX 


Abstraction  148, 166 ;  psychological 
account  of  190 

Adjectives,  and  abstraction  148; 
compound  61,  64;  and  mathe- 
matical concepts  140;  nature  of 
xiii 

Agreement,  figure  of  223,  228 ;  il- 
lustrations of  231 ;  Mill's  method 
of  118,  217,  242 

Algebra,  and  functional  deduction 
124,  130;  and  logical  principles 

135 
Algebraical  dimensions  185;  proof 

201 
Alphabet  and  numerical  notation 

Alternative  relation  of  propositions 

211 
Analogy,  a  criterion  of  certitude  250 
"And",  conjunctive  63;  enumera- 

tive  62 
Antilogism  78;  for  demonst.  induc- 
tion 227  ;  for  syllogism  80,  87 
Applicative  principle    10,  27,   104, 
118,   123,   129;   in   mathematics 
132 
Aristotle's  doctrine  of  proprium  125 
Arithmetic,   and  logic     133;    and 

number  158 
Arithmetical  processes  181 
Assertion  and  the  proposition  xiv,65 
Assertoric  and  hypothetic  243 
Association  and  inference  3,  7 
Associative  Law  128 
Attention  190 

Axioms,  establishment  of  33,  201 ; 
geometrical  201 ;  of  mathema- 
tics 123;  and  necessary  inference 
126 


Boole's  symbolic  logic  136 

Boyle's  Law  107,  no 

Brackets,  function  of  53,  122,  129 

Cantor  128,  137,  176 

Carroll,  Lewis  'j'] 

Categories,  definition  of  15;  and 
latent  form  55,  60,  139;  and 
magnitude  154 

Causal  formula  246 

Causation,  Law  of  218 

Cause  and  effect,  and  figures  of  in- 
duction 232 ;  and  absolute  mea- 
surement 179;  reversibility  107, 
116 

Certitude,  criteria  of  249 ;  demon- 
strative 250;  experiential  and 
rational  242;  of  hypothetical  pro- 
positions 244 ;  of  intuitive  gene- 
ralisations 192 

Characterisation,  a  relational  predi- 
cation 142 

Classes,  "comprising"  items  146, 
167  ;  and  genuine  constructs  62  ; 
and  extensional  wholes  166 ;  and 
number  154;  and  series  155; 
and  syllogism  87 

Class-names  and  symbolic  variables 

60 

Class-terms  and  syllogism  79,  84 

Combination  and  composition  236 

Commutative  Law  128 

Composite  propositions  and  demon- 
strative induction  212 

Composition,  and  combination  236 ; 
figure  of  222,  224,  228;  illustra- 
tions of  238, 248 ;  principle  under- 
lying 248 

Compounds,  nature  of  61 


H 


I 


f' 


n 


Comprising,  and   classes    146;    a 

relational  predication  142 
Conjunctional  functions  55,  62,  72 
Connectional  functions  54,  57,  141 
Connotation  and  property  125 
Constants,    absolute    and    relative 
120;    formal    and    material     43, 
141 ;  implicit  and  explicit  53 
Constitutive  condition  of  inference 

8,  10 
Constructs,  fictitious    61,  64;   and 
functions   48;   simple  and  com- 
pound 141 
Continuants  xi,  no 
Conversion  31,  39;  a  type  of  intui- 
tion 195  ;  relative  100 
Correlation,  factual  and   factitious 
1 56?   159;    functional    160;    one- 
one  158 
Counter-applicative  principle  28 
Counter-implicative   principle    29; 

relation  2n 
Counter-principles  of  inference  28 
Counting,    analysis    of    act     157; 
logical      principles      underlying 

158 

Co-variation,  in  economics  115; 
formulae  establishing  249;  and 
inductive  figures  218,  219,  229; 
law  of  106;  in  physics  n3 

Deduction  104;  functional  129; 
and  observation  n9;  range  of 
189,  213;  and  method  of  Resi- 
dues n8 ;  employment  in  Science 
216 

Demonstrative  induction  210;  cer- 
titude of  250;  figures  of  222, 227 ; 
Mill's  methods  217,  222;  use  in 
Science  216 

Demonstrative  inference  33,  102, 
132;  and  deduction  241;  and 
problematic  inference  132,  189, 
241 

Dependence,  concept  of  219 


Determinables,  and  categories  19; 
in  demonst.  induction  215;  and 
determinates  43,  62,  149,  195 ; 
and  distensive  magnitudes  169; 
and  intensive  magnitudes  172 

Difference,  figure  of  222,  227 ;  illus- 
tration of  228;  Mill's  method  of 
118;  principle  underlying  247 

Disjunctive  propositions  2n  ;  prin- 
ciple, and  the  syllogism  78 

Distensive    magnitudes    162,    168, 

173 
Distribution    89,    198;    syllogistic 

rules  of  92 
Distributive  Law  128 
Division,  concrete  183;  contrasted 

with  addition  181,  188 

Enthymeme  100 

Epistemic  condition  of  inference  8 ; 
nature  of  term  "hypothesis"  242 

Equality,  measurement  of  178;  nu- 
merical 145,  149,  159 

Equations,  connectional  n2;  func- 
tional 126;  limiting  127;  linear 
107,  w-j 

Ethical  judgments  and  intuition 
194 

EucHd  201,  204 

Experiential  certification  36 

Experimentation,  rule  for  220;  con- 
ditions for  valid  249 

Extension,  applications  of  term  166; 
a  species  of  magnitude  166,  174 

Factitious  correlations  156,  158 
Factual  and  factitious  correlation 

156,  159 

Fallacies,  material  and  formal  loi 

Fechner's  "just  perceptible  differ- 
ence" 170 

Figures  of  induction  221  ;  illustra- 
tions of  228;  use  of  232 

Figures  of  syllogism  77,  87;  dicta 
for  first  three  80,  83 ;  fourth  87 


256 


INDEX 


INDEX 


Form,  of  argument  208 ;  elements 
of  53;  and  matter  191;  and  pri- 
mitive ideas  138 

Formal  correlation  160;  and  ma- 
terial 139 ;  relations,  table  of  144 

Formulae,  establishment  of  33, 127, 
129,  195;  of  functional  induction 
249;  range  of  129,  131 

Functional  conjunction  237  ;  corre- 
lation 160;  deduction  124;  in- 
duction 246;  syllogism  103,  106, 
120,  127 

Functions,  conjunctional  72 ;  con- 
nected and  disconnected  130; 
and  constructs  48,  130;  descrip- 
tive 69;  formal  and  non-formal 
5O1  75;  propositional  71;  and 
variants  49,  57;  varieties  of  55, 
66,68 

Geometrical  figures,  use  of  201,203 ; 
abuse  of  206 

Geometrical  induction  197,  205 ; 
magnitudes  187;  proof  201,  204 

Geometry,  analytical  204 ;  and 
functional  deduction  124;  Mill 
on  foundations  of  191 

Gravitation,  an  instance  of  function- 
al syllogism  109;  probability  of 
formula  250 

Grounds  of  argument  38 

Hume's  philosophy  82 
Hypothetical  propositions  11,  242; 
and  problematic  244 

Identity,  of  adjectives  149;  rela- 
tion of  20,  142 

Illustrations,  choice  for  syllogism 
77,  81,  loi ;  of  demonstrative  in- 
duction 212,  213,  215,  216;  of 
summary  induction  197,  198 

Illustrative  symbols  41,  46 

Imagery,  and  geometrical  induction 
202 ;  and  intuited  universals  193 


I mplication, and  demonst.  induction 
210;  and  hypotheses  243;  rela- 
tion to  inference  xv,  i,  76 ;  a  rela- 
tional predication  142 
Implicative  formula  152;  principle 

10,  27,  104,  118;  relation  211 
Including,  and  extensional  wholes 
167;  a  relational  predication  142 
Independence,   notional   and   con- 

nectional  108 
Induction,    relation   to    Deduction 
189,213,243;  demonstrative  189, 
210,   227;    figures   of   221;    and 
functional  formulae  105,  131 ;  in- 
tuitive   29,    189;    mathematical 
132,   133;   and  observation   119; 
pre-scientific    219;    problematic 
189,  216,  219,  240;  type  of  Pro- 
position   underlying    66 ;     pure 
240;  summary  or  perfect  197 
Inductive  principle  23,  38 
Inference,  and  implication    i,  76, 
152;    paradox  of   10,   136;    pre- 
requisites of  2 :  principles  of  10 ; 
psychological    conditions    of    4; 
conditions  for  validity  7 
Infinity,  and  cardinal  numbers  161 ; 
orders  of  128;  transfinite  aggre- 
gates 155,  160 
Instantial  premiss  210,  216 
Integers,  finite  133,  161 ;  notion  of 

139,  154;  odd  and  even  161 
Intensity  and  reality  172 
Intuition,  and  experience    191 ;   in 
inference  31,  33;  and  sensation 
192;  of  space  202;  and  syllogism 
90 
Intuitive  induction    29,    189;    and 
certitude    192 ;    experiential  and 
formal  192;  and  logical  formulae 
195 ;   involved  in  geometry  205  ; 
distinguished  from  summary  200 

Jevons,   Elementary  Lessons    116, 
125;  on  induction  244 


li 


'1       > 


^ 


?^ 


Kant's  views  on  geometry  202; 
philosophy  82 

Keynes,  J.  M.,  Treatise  on  Proba- 
bility 253 

Language  and  symbolism  44 

Laws  of  Nature  106,  126 

Logic,  relation  to  mathematics 
123,  132,  137,  141;  relation  to 
science  216,  228,  231,  235;  sym- 
bolic 136 


257 


Magnitudes,  absolute  and  relative 
205 ;  abstract  and  concrete   161, 
181;  comparison  of  174;  disten- 
sive  168;  etymology  of  153;  ex- 
tensive 162;  intensive  172;  and 
material   variables    144;    simple 
and  compound  180;  varieties  of 
150,  162,  187 
Major  term  76;  rules  for  94 
Mathematical  induction  133;  sym- 
bolism 136,  141 

Mathematics,  and  functional  formu- 
lae 105,112,120,126;  and  relation 
to  logic  123,  137,  141,  151;  and 
principles  of  inference  132,  152 
easurement,  of  extensive  magni- 
tudes 175;  of  geometrical  mag- 
.  nitudes  187 

fiddle  term  ^^ ;  rules  for  93 
Mill,  J.  S.,  on  foundations  of  geo- 
/  metry  191, 208;  inductive  methods 
I  217,  229,  332  ;  inductive  methods 
'  criticized  217,233,241;  on  perfect 
induction    197;     on    probability 
value    251;    definition  of  "pr/^- 
prium"  125;  method  of  Residie^  ' 
116,  118,  222;  on  syllogism  xvii,  ' 

j  244  

Minor  term  76;  rules  for  Q4    J*-  •  i 

Air  r't       ,    ,  •    . 

^Inemonic  verses  97  '    •  * 

IVjloods  of  syllogism   76,  84 ;    rulc^ 
for  valid  86  ;        : 

ultiplication,  concrete  181,  con- 
trasted with  addition  181,  188 


Number,  alphabetical  notation  of 
158;  cardinal  and  ordinal  155, 
161;  and  classes  154;  psycho- 
logical aspect  of  155 

Obversion  91,  99 

Occurrents  xi 

Operators,  logical  status  of  141; 
and  number  158 

"Or,"  function  in  genuine  con- 
structs 63 

Order,  serial  and  temporal  157 


Particulars  and  universals  191,  192 

Peano  137 

Per^  meaning  of  183 

Perception,  analysis  of    190;    and 

inference  5 
Petitio principii  xvii,  10,  136 
Postulates,  of  problematic  induction 

189,  240;  of  science  219 
Predesignations  and  functions  69 
Predicational  functions  56,  72 
Premisses,  composite   210;   in  in- 
ductive  figures    218;     instantial 
210,  216;   subminor  and  super- 
major  21;  of  syllogism  76 
Principia  Mathematica  66,  138 
Principles,  enumeration  of  32 ;  epi- 
stemic  character  of  31 ;  function 
of  23 ;  of  inference  10 ;  underly- 
ing inductive   figures    247,  248; 
underlying  mathematics  123,  158 
Principles  of  Mathematics  xiii,  155, 

161,  165 
Probability,  conditions  for  high  de- 
;       ^ree.of  251  ;  law  of  error  253 
'    Piotleniati.:   in^u^tbn,  and   func- 
tional 246';  and  prescientific  in- 
.^     .  vert'gation  .216^  319,  220,  240 
•.  Problematic  ^iofeteni-j^,  and  demon- 
strative 132;  1^9,218;  andhypo- 
,,,^  thetica!    244;   and  summary  in- 
*»**  dutticn  iq8i'2O0' 
Proof,  analytickl  and   geometrical 
201 ;  science  of  200 


258 


INDEX 


Proper  names  and  numbers  156 
Property,  notion  of  125 
Propositional    functions     66,    71  ; 

types  66 
Propositions,    and   assertion     xiv; 

composite  210;  structural  14 
Psychological  account  of  inference 

4 ;  account  of  symbolism  44 

Quantity,  relation  to  magnitude  162 

Ratios,  and  addenda  171 ;  and  an- 
gles 186;  notion  of  139 

Relational  predications  142;  many- 
one  145;  many-many  156;  one- 
one  158 

Relations,  adjectival  nature  of  xii ; 
extensional  treatment  of  xii,  159 

Residues,  Herschel's  method  of 
118,  222,  249;  Mill's  method  of 

116 
Resolution,  figure  of  222,  226,  228 ; 

illustration  of  239 
Reversibility,  principle  of  107,  1 16 
Russell  B.,  principle  of  abstraction 
146;  notion  of  class  148;  on 
equality  146,  I59,  i75;  notion  of 
function  52,  66;  on  symbolism 
138;  on  time  and  space  165; 
theory  of  types  73 


Stretches,    quantitative    measure- 
ment of  178;  varieties  of  163 
Structural  propositions  14 
Substantive,  compound  61 ;  nature 

of  xi 
Subsumption  103,  120,  124 
Summary  induction  200 
Supernumerary  moods  85,  88 
Syllogism,  analysis  of  12,  17,  76; 
dicta  for  figures  80, 83 ;  functional 
103,  120,  127;  illustrations  of  77» 
81,  loi ;  importance  of  102  ;  and 
mathematics  123;  Mill's  analysis 
xvii;   principle   of  21,  24;    and 
summary    induction     197;    and 
thought  process    100;    rules  for 
valid  moods  89 
Symbolism,  use  in  inductive  figures 
234;  mathematical  136, 141 ;  and 
meaning   45;   psychological  ac- 
count of  44 ;  value  of  39, 41 » 136 ; 
varieties  of  41,  129 

Ties,  nature  of  53;  temporal  aj 

spatial  164 
Time  and  space,  logical  nature 

163;  measurement  of  176;  re\ 

tivity  of  165 


Science,   and   demonst.   induction 
216;  and  inductive  figures  228, 
231,  235  ;  postulate  of  219 
Sensational  magnitude  170,  180 
Sense-data  and  induction  38 
Sense-experience, andinti^U.ioo  192 ;  . 

nature  of  i^/',  •   r:  :  '  !  !    !  ^    | 
Sentence  and  projJositroii  5^ 
Simple  enumenatipn.2a.8  ^, .  , ,.  .^     , 
Simplicity,  a%Crittrion*&/'.crrtitiide; 

250 
Sorites  97       .^    .    ,..      •       ••!•"" 
Space,  Euclidian  a>4  nok-EUolicJ- • 

ian   201  ;   me^siiVemdnt  "of'  176*,*  * 

relativity  of  165 


Universal  propositions  11 
Universalisation,   formula  of   21 

220,  222 
Universals,  apprehension  of  191 

Variables,  apparent  58,66;  in  fun 
tional  formulae  108, 1 12,  120, 12 
.  k  .1 3iP I  formal  and  material  1 40,  i 
A^sGriants  71 
Verbdf  propositions  125 

.Verification  119 


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.  dugtion  199 

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'   and  extensional  166 ;  in  geomet  ry 

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PRINTED  IN  ENGLAND 

AT  THE  CAMBRIDGE  UNIVERSITY  PRESS 

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LOGIC 


PART  III 


"V^* 

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LOGIC 

PART  III 

HE  LOGICAL  FOUNDATIONS  OF  SCIENCE 


CAMBRIDGE  UNIVERSITY  PRESS 

C.  F.  CLAY,  Manager 

LONDON   :  FETTER  LANE,  E.C.4 


NEW  YORK  :  THE  MACMILLAN  CO. 

BOMBAY 

CALCUTTA  -  MACMILLAN  AND  CO.,  Ltd. 

MADRAS 

TORONTO   :  THE  MACMILLAN  CO.  OF 

CANADA,  Ltd. 
TOKYO :  MARUZEN-KABUSHIKI-KAISHA 


BY 


W.  E.  JOHNSON,  M.A.,  F.B.A. 

HON.  D.LITT.  MANCHESTER, 

FELLOW  OF  king's  COLLEGE,  CAMBRIDGE, 

SIDGWICK  LECTURER  IN  MORAL  SCIENCE  IN  THE 

UNIVERSITY  OF  CAMBRIDGE 


ALL  RIGHTS  RESERVED 


CAMBRIDGE 
AT  THE  UNIVERSITY  PRESS 

1924 


m 


^L^  ^n^l.   Li^^[  Cr^) 


V  3 


PRINTED  IN  GREAT  BRITAIN 


A 


CONTENTS 


INTRODUCTION 

PAGE 

§  I  j  Review  of  the  lines  of  discussion  occupying  Parts  I  and  II.    Pre-mathe- 

matical  Logic xiii 

§  2i  Summary  of  the  author's  treatment  of  Mathematical  Logic  .         .         .      xiv 

§  3i.  The  ontological  topics  introduced  in  Part  III :  comparison  with  the 
treatment  accorded  to  these  topics  by  other  logicians.  The  place  of 
postulates  in  inductive  theory     ........      xvi 

§  41  The  division  of  existents  into  continuants  and  occurrents     .         .         .    xviii 

§5^;  The  dualistic  view.    Meanings  of  *  of xix 

§  6]  Examination  of  the  distinction  between  occurrent  and  event :  the  former 
being  identified  and  discriminated  by  reference  to  difference  of  adjec- 
tival determinable,  the  latter  by  reference  to  difference  of  location        .      xxi 

§7.  Agent  and  Patient.   Immanent  and  Transeunt.    Monadism  and  Monism   xxiii 
§  8.  Combination  of  Immanent  with  Transeunt  Causation  ....     xxv 
§9.  Interaction  between  *mind'  and  'body.'   Parallelism  and  Correspon- 
dence.   Parallelism  and  denial  of  Causality.    Psychical  and  Physical 
processes  presented  in  cycles xxvi 

§  10.  One-sided  correspondence.    Impartial  Dualism xxviii 

§11.  Attempt  to  meet  attacks  upon  Impartial  Dualism        ....    xxxi 

§12.  Invariability  and  Causality xxxii 

§13.  Misrepresentations  of  the  deterministic  position xxxiii 

§  14.  Explanation  of  the  necessity  for  introducing  the  discussions  of  psycho- 
logical and  metaphysical  topics  into  Logic xxxv 


n^, 


CHAPTER  I 


FACT  AND  LAW 


§  I.   Statements  of  fact 

§  2^  Reduction  of  statements  of  fact  to  the  form  :  A  certain  Pis  p 

§  3.f  Verbal  expression  for  the  distinction  between  the  universal  of  fact  and 
the  universal  of  law 

§  4  J  Resum^  of  uses  of  the  term  *  possible ' 

§  5.    Inadequacy  of  the  purely  factual  interpretation  of  certain  types  of  pro- 
position      

§6|.  The 'possible' and  the 'hypothetically  necessary'       .         .         .         . 

^3 


I 
2 

4 
6 

II 
14 


VI 


CONTENTS 


CONTENTS 


vu 


PAGE 


CHAPTER  II 

THE  CRITERIA  OF  PROBLEMATIC  INDUCTION 

lge 

§1.  Instantial  propositions  as  premisses  of  induction          .         .         .         •  I    '^ 

§2.  The  *  course' of  nature  and  the 'laws' of  nature.         .         .         .  l    '7 

§  3.  Complementary  enumeratives  and  complementary  universals       .         .  1   '9 

§4.   Requirement  of  maximum  variety  amongst  examined  instances    .         •  I  '^ 

§  5.  Combination  of  variety  and  similarity I  24 

§  6.  Number  and  proximity  as  substitutes  for  variety  and  similarity    .         •  I  ^4 

§  7.  Independence  of  characters  and  incongruence  of  instances   .         .         •  I  ^5 

§  8.  First  formulation  of  the  criterion  for  the  Method  of  Complementaries .  I  ^7 

§  9.   Criterion  of  precision ^^7 

§  10.  Complex  comprehensive  exactitude 

§11.  Tabular  schematisation  corresponding  to  Bacon  and  Mill's  empirical 

generalisation    .         .         .         .         .         .         .         .         .         .         .  f  28 

§12.   Discussion  of  the  relation  between  'hypothesis'  and  'generalisation'    .  f  30 
§  1 3.  The  degree  of  probability  varies  directly  with  the  degree  of  ascertained 

accordance 1  ^^ 

CHAPTER  III 

1 

DEPENDENCY  AND  INDEPENDENCY  ' 


§  I.  Separation  in  thought  between  the  determining  and  the  determined 
characters — these  being  presented  as  merely  conjoined  in  fact 

§  2.  In  experimentation  the  determining  factors  are  known  de/ore  and  the 
dependent  factors  only  ajier  the  result  of  the  experiment     . 

§  3.  Errors  in  observation  (unaided  by  experiment)  resulting  from  the  false 
supposition  that  certain  factors  are  independent  which  are  in  reality 
dependent         


CHAPTER  IV 


EDUCTION 

§  I.  Inference  (so-called)  from  particulars  to  particulars  should  be  termed 
eduction     ............ 

§  2.  Eduction  involves  a  minor,  middle  and  major  premiss  containing  not 
only  a  minor  and  a  major  term  but  also  two  middle  terms  respectively 
intensional  and  extensional.    Schematisations  of  eduction    . 

§  3.  The  commonly  alleged  distinction  between  induction  and  analogy  should 
be  replaced  by  the  distinction  between  the  two  mediating  terms  respec- 
tively extensional  and  intensional  required  for  eduction  and  for  generali- 
sation.   Cases  tabulated  where  all  the  evidential  data  are  in  favour  of 
a  certain  educed  conclusion        .....         ... 


36 
38 


it 
4 


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43 


44 


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46 


§  4.  In  estimating  the  evidential  force  of  instantial  data,  characters  must  be 
counted  only  so  far  as  they  constitute  an  independency,  and  instances 
only  so  far  as  they  constitute  a  variancy 

§  5.   Further  explication  of  the  above  principle 

§  6.  Further  exposition  of  the  principle  as  regards  evidential  premisses  with- 
out respect  to  inferred  conclusion 

§  7.  The  nature  of  the  inferred  conclusion  in  its  relation  to  the  evidential 
premisses 

§8.  Irreducible  contrast  between 'subject' and 'predicate' 

§  9.  The  above  contrast  as  corresponding  to  the  fundamental  distinction 
between  separation  and  discrimination 


CHAPTER  V 
PLURALITY  OF  CAUSES  AND  OF  EFFECTS 

§  I.  Incomplete  assignment  of  Cause  yields  alternative  Effects  just  as  in- 
complete assignment  of  Effect  yields  alternative  Causes 

§  2.  An  assignment  of  Cause  or  of  Effect  is  complete  only  in  reference  to  the 
Effect  or  to  the  Cause 

§  3.  A  conjunction  of  cause-factors  constitutes  a  completed  cause ;  a  con- 
junction of  effect-factors  constitutes  a  completed  effect 

§  4.  Plurality  means  that  some  but  not  all  of  the  possible  values  of  a  deter- 
minable may  be  put  into  the  subject-terms  of  the  universals  in  which 
the  same  determinate  character  is  predicated       ..... 

§  5.  Correction  of  the  demonstrative  figures  of  induction  required  by  the 
consideration  of  a  possible  plurality  of  causes  or  of  effects    . 

§6.  Reversibility  of  the  universal  proposition  connecting  the  completed 
cause  and  the  completed  effect 

J  7.  The  step-by-step  process  by  which  the  completed  cause  and  the  com- 
)leted  effect  are  reached 


CHAPTER  VI 

CAUSE-FACTORS 

Preliminary  exposition  of  the  notion  of  change  in  its  reference  to  a 
Continuant 

Continuant  as  Cause  must  be  distinguished  from  Occur  rent  as  Cause. 
Cause  and  Effect  are  coordinate  when  interpreted  as  Events  and  each 
is  reciprocally  inferable  from  the  other.  Nevertheless  the  temporal  rela- 
tion is  r^arded  as  not  reciprocal         . 

Alleged  distinction  between  two  types  of  objective  law 

The  properties  of  continuants  regarded  as  causal  .... 

Effects — like  causes — not  merely  resolvable  into  events  or  occurrences 

Temporal  sequence  of  cause-occurrence  and  effect-occurrence  explained 
in  reply  to  philosophical  criticism 


48 
48 

50 
53 


54 
56 

57 

59 
60 

62 
63 


66 


68 
70 

71 

72 

74 


I 


VIU 


CONTENTS 


CHAPTER  VII 


THE  CONTINUANT 


PAGE 


§  I.  The  fundamental  notion  of  causal  connection  between  movements  that 
occur  in  space  entails  reference  to  a  physical  continuant  conceived  as 
that  which  moves        ..........       78 

§  1.  The  term  'continuant'  is  chosen  to  replace  'substance'  in  order  to  free 
the  notion  from  certain  philosophical  implications  inseparable  from  the 
latter,  and  to  emphasize  the  inevitable  residuum  which  (as  maintained 
by  the  writer)  is  indispensable  for  science.  Thus,  in  the  first  place,  the 
extension  of  continuance  to  an  infinite  future  and  an  infinite  past  which 
is  often  attributed  by  philosophers  to  substance  may  be  dispensed  with 
in  the  scientifically  conceived  continuant    ...... 


§  3.  Secondly,  substantival  continuance  does  not  necessarily  entail  any 
adjectival  changelessness 

§  4.  Thirdly,  the  ultimate  substantival  continuant  is  not  necessarily  simple. 
Moreover,  the  existential  components  which  may  constitute  a  continuant- 
unity  may  not  themselves  be  continuant      ...... 

§  5.  What  in  general  holds  of  the  physical  continuant  holds  also  of  the 
psychical  continuant ;  but  the  structure  of  the  latter  exhibits  far  higher 
complexity  than  the  former.  For,  whereas  motion  is  the  sole  funda- 
mental mode  of  manifestation  of  the  physical  continuant,  there  are  many 
irreducible  but  interconnected  modes  of  manifestation  to  be  attributed 
to  the  psychical  continuant.  Within  each  of  these  several  modes  the 
conception  of  change  must  be  separately  applied  .... 

§  6.  The  conception  of  change,  moreover,  involves  the  replacement  of  one 
by  another  manifestation,  of  which  different  determinate  characters 
under  the  same  determinable  may  be  predicated 

§  7.  The  mutually  implied  conceptions  of  substance  and  causality  (in  their^ 
residual  scientific  significance)  lead  to  the  notion  of  property ^  which  is* 
the  appropriate  adjective  characterising  a  continuant  as  contrasted  witl 
an  occurrent.   A  property  must  be  conceived  as  a  defined  potentiality; 
other  adjectives  are  conceived  as  descriptive  of  actualities   . 

§8.  The  comparatively  primitive  attempts  to  systematise  the  manifold  ol 
reality  illustrate  the  same  principles  and  postulates  which  govern  the ' 
procedure  of  advancing  science 

§  9.   Sub-continuants  and  sub-occupants 

§10.  Causality  within  the  manifestations  of  a  single  continuant    . 

§11.   Uniformities  embracing  ^{^r^w^  continuants       ,         ♦         .         .         . 

§  12.  The  unity  of  a  continuant  exhibited  in  causal  formulae.   Alterable  and 
unalterable  properties 

§  13.  Comparison  of  the  views  here  maintained  with  those  of  Kant      .         ^ 

§  14.  Fundamental  contrast  between  the  author's  views  and  those  widely 
current  since  Hume  and  at  the  present  day 


80 
80 

81 


82 


84 


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86 


4. 


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CONTENTS 


CHAPTER  VIII 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND 


IX 


PAGE 


§  I .  The  effect  of  purposive  control  in  modifying  the  more  mechanical  mental 
processes  points  to  a  form  of  causality — operating  within  the  experience 
of  a  single  individual— which  is  closely  analogous  to  the  form  of  cau- 
sality termed  transeunt 10* 

§  «.  If  there  is  any  direct  determining  influence  of  the  psychical  upon  the 
physiological  or  conversely,  such  influence  undoubtedly  comes  under 
the  head  of  transeunt  causality.  If,  for  example,  we  assume  that  sensa- 
tions and  other  quasi-mechanical  mental  processes  are  directly  deter- 
mined by  neural  processes,  there  will  be  a  direct  correlation  between 
such  mental  processes  on  the  one  hand  and  the  neural  processes  on  the 
other.  If,  fiirther,  we  assume  that  the  phases  of  feeling  and  cognition 
are  partially  determined  by  sensations,  then  there  will  be  a  partial — 
but  indirect — correlation  between  the  former  and  the  latter.  But  such 
correlation  must  be  limited  to  those  variations  which  can  be  said  to 
correspond  with  one  another.  And  it  would  appear  that  there  are  no 
variations  in  neural  process  which  could  correspond  to  the  variations  of 
feeling  and  cognition '©4 

§  3.  Any  felt  effort  or  strain  that  may  be  incurred  by  mental  activity  has 
both  a  physiological  and  a  sensational  aspect  which  can  properly  be 
said  to  correspond.  But  so  far  as  the  effects  of  such  effort  are  intended, 
we  have  reasonable  ground  for  asserting  the  operation  of  transeunt 
causality,  in  which  the  causal  agency  is  psychical  and  the  direct  effect 
physiological      .         .         .         .         .         •         •         •         •         •         .107 

§  4.  The  presumption  that  foreknowledge  operates  in  the  psychical  deter- 
mination of  physical  effects,  is  tantamount  to  attributing  real  causal 
efficiency  to  such  foreknowledge.  But  if  this  foreknowledge  could  be 
reduced  to  merely  physiological  terms,  mental  causality  would  be  an 
illusion •     '®9 

§  5.  Mental  activity  assumes  two  distinct  types,  (i)  Motor  activity  which 
produces  sensational  effects,  and  (2)  The  activity  of  attention  which 
produces  cognitional  effects.  In  both  cases,  grounds  are  put  forward 
for  maintaining  that  the  causal  agency  entailed  is  purely  psychical       .     no 

§  6.  Explicit  grounds  for  the  contention  that  changes  of  cognitive  phase  have 

no  counterparts  in  the  changes  of  neural  process  .         .         .         •     "S 

§  7.  The  prevalent  confusion  between  images  and  ideas  largely  accounts  for 

the  refusal  by  physiologists  and  psychologists  to  accept  the  above  view     116 

§  8.  An  analysis  of  the  phases  of  consciousness  accompanying  reflex  processes 
best  illustrates  the  contrast  between  physiological  and  psychical  causal 
agency '^7 

§  9.  But  an  analysis  of  conative  conflict  yields  the  most  important  indication 

that  psychical  agency  is  really  causally  operative  .         .         .         .     120 

§  10.  The  cognitive  aspects  of  deliberative  process  complete  the  grounds  for 

our  main  contention '  ^4 

I II.  'Judgments  of  value'  stand  to  'conations'  as  cause  to  effect,  not  con- 
versely              .....     125 


CONTENTS 


*» 
1 


PAGE 


117 


128 


119 


131 


134 


CHAPTER  IX 

TRANSEUNT  AND  IMMANENT  CAUSALITY 

§  I.  The  event  termed  'Movement'  cannot  be  reduced  to  merely  spatio- 
temporal  terms;  since  it  requires  something  that  moves,  and  that 
retains  its  continued  identity  within  the  spatio-temporal  bounds  of  the 
event         ...•...••••• 

§  2.  The  causality  formulated  in  the  first  law  of  motion  is  wholly  immanent, 
but  that  formulated  in  other  dynamic  laws  is  essentially  transeunt 

§  3.  Analysis  of  the  immanent  and  transeunt  factors  entering  into  an  ele- 
mentary physical  process 

§  4.  Contrast  to  the  previous  illustration,  where  cause  and  effect  are  reversed. 
In  cases  of  immanent  process,  where  a  cause-occurrent  and  an  effect- 
occurrent  are  simultaneous,  our  ground  for  deciding  which  of  the  two 
occurrents  is  cause  and  which  is  effect  is  based  upon  the  principle  that 
that  occurrent  which  is  effect  in  the  transeunt  process  is  cause  in  the 
immanent  process      .,.....••• 

§  5.  In  the  analysis  of  emotional  experiences,  which  entail  diflFused  organic 
sensations,  another  illustration  is  afforded  of  the  ways  in  which  transeunt 
and  immanent  causality  are  distinguished  and  combined 

§  6.  Fundamental  distinctions  between  the  psychical  and  the  physical  con- 
tinuant     .         •        . 13^ 

§  7.  The  finally  unique  distinction  between  the  two 139 

§  8.  Formulae  which  are  correctly  expressed  in  terms  of  immanent  causality 
as  regards  unitary  wholes  are  frequently  equally  correctly  and  more 
adequately  expressed  in  terms  also  of  transeunt  causality  as  regards 
constituent  parts         .         .  .         .  .         •         •         •  •         .140 


CHAPTER  X 
CONVERGENT  AND  DIVERGENT  CAUSALITY 

§  I.  Diagrammatic  representation,  by  the  use  of  parallel,  converging  and 
diverging  lines,  to  explain  the  different  forms  assumed  in  causal 
complexes, — parallel  lines  being  employed  to  represent  causal  inde- 
pendence  143 

§2.  Application  of  above  to  dynamic  and  chemical  formulae      .        .         .     146 

§  3.   Further  application  to  psycho-physical  formulae  .         .         .         .148 

§  4.  Extension  of  the  diagrams  to  illustrate  the  more  complicated  formulae 

in  physics .         . 152 

§  5.  Similar  exposition  of  the  more  complicated  forms  of  psycho-physical 

causality 156 


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CONTENTS  xi 

CHAPTER  XI 

TEMPORAL  AND  SPATIAL  RELATIONS  INVOLVED 

IN  CAUSALITY 

PAGE 

§  I.  The  conception  of  connectional  determination   as  involving  spatio- 
temporal  relations •         •         •         •         .101 

§2.   Order:  discrete  and  continuous  ...••••     i"^ 

§  3.   The  possibility  of  discontinuous  change ^"5 

§4.  The   idea  of  connectional   determination   extended  to  interpsychical 

causality 109 

§  5.   Analogies  between  the  spatially  inner  and  outer  on  the  one  hand  and 

the  temporally  prior  and  posterior  on  the  other i7' 

§6.   Explanation  of 'potential' causality '73 

§7.   Examination  of  certain  elementary  physical  processes  .         .         .     i75 


APPENDIX  ON  EDUCTION 

§  I .  Notation  to  be  adopted.    Two  elementary  formulae     .         •         •         .  1 78 
§  2.  The  notion  of  probability.    Proposal  and  supposal.   Contrast  with  the 

relation  of  implication        .         . '79 

§3.  The  two  working  axioms  of  the  probability-calculus    .         .         .         .181 

§  4.  Four  corollaries  from  the  two  axioms 'o' 

§  5.  Necessity  for  special  postulates  before  the  theorems  of  the  calculus  can 

be  applied  to  any  actually  given  problem '82 

§  6.  Two  postulates  are  adopted  in  the  proposed  establishment  of  a  theory 

of  eduction         .         .         .         .         •         •         •         •.         •         •         '183 

§7.  Formal  proof  of  the  eductive  theorem 184 

§8.  Elucidation  of  the  formula  for  successive  values  of  N.    Mnemonic 

schematisation  .....••••••  '°^ 

§  9.   Discussion  of  the  grounds  on  which  the  adoption  of  the  two  postulates 

is  based  and  of  the  type  of  case  for  which  the  postulates  are  legitimate  187 

INDEX '90 


{ 


I     4. 


4 

4 


4 


i 


♦s- 


4 


INTRODUCTION 

§  I.  The  subjects  discussed  in  Parts  I  and  II  come 
within  the  scope  of  what  may  be  called  Formal  Logic. 
Here  the  proposition  is  taken  to  be  the  immediate  ob- 
ject of  a  possible  assertion ;  and  a  consideration  of  its 
nature  leads  to  the  conception  of  the  antithesis  and  con- 
nection of  substantive  with  adjective,  as  disclosed  in 
the  analysis  of  the  simplest  articulate  form  of  judgment. 
The  function  of  language  and  more  particularly  of  names 
is  examined.  It  is  held  that  the  different  forms  assumed 
by  compound  propositions  are  indicated  by  various  words, 
not  standing  for  substantival  or  adjectival  constituents, 
but  expressive  of  the  modes  in  which  simple  propositions 
or  their  constituents  are  to  be  connected  by  constructive 
thought.  Such  considerations  lead  to  a  preliminary 
definition  and  enumeration  of  logical  categories  roughly 
corresponding  to  (and  replacing)  the  grammatical  enu- 
meration of  parts  of  speech. 

In  the  more  detailed  examination  which  follows, 
substantives  proper  or  existents  are  distinguished  from 
quasi-substantives,  adjectives  predicable  of  the  former 
being  termed  primary  and  those  predicable  of  the  latter 
secondary.  Modality,  in  its  formal  aspects,  is  treated 
under  the  more  general  heading  of  secondary  proposi- 
tions. Adjectives  are  divided  into  transitive  adjectives 
(otherwise  relations)  and  intransitive  adjectives,  in  pre- 
cise analogy  with  the  grammatical  division  of  verbs : 
and  again  into  monadic,  dyadic,  triadic,  etc.  according 
to  the  number  of  substantive-terms  which  are  entailed 


XIV 


INTRODUCTION 


INTRODUCTION 


XV 


in  their  employment.  A  prominent  place  is  given  to  the 
distinction  and  connection,  amongst  adjectives  in  general, 
between  adjectival  determinates  and  adjectival  deter- 
minates. This  distinction  is  utilised  in  all  the  further 
developments  of  logical  theory.  The  relations  between 
inference  and  implication,  the  former  of  which  is  essen- 
tially epistemic  and  the  latter  essentially  constitutive  are 
entered  into  at  considerable  length.  In  particular, certain 
general  and  {und3.ment3\  prznczp/es  of  inference  are  laid 
down  and  contrasted  as  formal  with  the  premisses  of 
inference  which  are  material. 

Inferences  and  implications  are  divided  into  the  two 
species  demonstrative  and  problematic.  The  term  in- 
duction has  been  used,  with  some  hesitation,  to  include 
four  species — intuitive,  summary,  demonstrative  and 
problematic.  The  first  three  of  these  are  examined  in 
Part  II,  the  fourth  being  reserved  for  Part  III.  Deduc- 
tive inference  or  implication  is  treated  in  connection 
with  the  intuitive  foundations  of  pure  logic  and  pure 
mathematics;  as  also  with  summary  induction. 

§  2.  It  is  contended,  in  agreement  with  most  recent 
logicians,  that  Arithmetic  and  (more  generally)  Pure 
Mathematics  develops  from  Pure  or  Formal  Logic:  i.e. 
that  the  conceptions  and  axioms  underlying  the  former 
are  none  other  than  those  underlying  the  latter.  If  any 
distinction  is  to  be  made  between  Pre-mathematical 
Logic  and  Pure  Mathematics  it  is  suggested  that  the 
latter  introduces  certain  adjectives  and  relations  which 
in  the  strictest  sense  are  constant,  i.e.  represented  by 
words  or  symbols  of  which  it  is  essential  for  the  science 
that  the  meanings  should  be  understood  in  one  invari- 
able sense;    whereas  the  intelligent   apprehension  of 


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pre-mathematical  formulae  requires  that  symbols  for 
adjectives  and  relations  in  general  should  be  understood 
merely  illustratively  to  stand  indifferently  for  any  actual 
adjectives  that  might  be  substituted  for  them. 

Now,  in  the  transition  from  pre-mathematical  to 
mathematical  logic,  the  first  notions  that  demand  ex- 
plicit recognition  are  those  of  identity  and  (its  contrary) 
otherness  or  diversity.  These  two  relations  are  applicable 
to  any  entities  whatsoever  coming  under  any  category 
whatever.  Thus  if  a  unambiguously  denotes  any  entity 
whatever  and  b  unambiguously  denotes  any  entity 
whatever,  then  (so  far)  the  entity  denoted  by  a  may  be 
identical  with  and  may  be  other  than  that  denoted  by 
b.  At  this  point,  the  two  axioms  that  identity  and 
otherness  are  co-alternate  and  co-disjunct  have  to  be 
explicitly  formulated.  Speaking  loosely,  the  relation  of 
identity  yields  the  notion  of  one  and  that  of  otherness 
yields  the  notion  of  two.  More  accurately  and  precisely! 
the  conception  of  number  is  developed  from  that  of  a 
certain  sub-division  of  the  genus  relation  termed  one-one', 
and  one-one  relations  are  defined  entirely  in  terms  of 
identity  and  otherness;  i.e.  no  other  notions  than  these 
are  involved  beyond  those  appertaining  to  pre-mathe- 
matical logic.  In  this  way,  the  definition,  not  only  of 
any  assigned  finite  number,  but  even  of  infinite  number 
introduces  (besides  pre-mathematical  notions)  identity 
and  otherness  alone.  In  the  higher  branches  of  arithmetic 
other  relations,  dyadic,  triadic,  etc.,  are  introduced,  espe- 
cially those  which  develop  from  the  general  notion  of 
order \  and  these  are  all  expressed  and  defined  in  terms 
of  words  or  symbols  having  a  fixed  invariable  meaning 
that  must  be  understood  by  the  mathematician  as  such. 


« 


p 


XVI 


INTRODUCTION 


Not  only  must  the  mathematician  understand  the 
meanings  of  the  constant  symbols  introduced  and 
defined  in  the  science,  but  also  his  intelligent  assent  is 
required  to  be  given  to  certain  axioms  (or  primarily 
fundamental  propositions)  expressed  in  terms  of  these 
symbols ;  and  his  intelligence  must  be  further  exercised 
in  following  the  demonstrative  procedure  by  which 
derivative  formulae  are  progressively  inferred.  He 
discovers,  not  only  the  comparatively  unimportant  fact 
that  the  conclusions  are  true  provided  that  the  originally 
premised  axioms  are  true,  but  also  the  more  important 
fact  that  the  conclusions  follow  demonstratively  from  a 
judicious  combination  of  these  axioms  and  these  alone 
— none  other  being  required.  The  account  of  symbolism 
and  allied  topics  in  Part  II  includes  references  to  pro- 
cesses of  thought  and  thus  is  largely  psychological — 
in  this  respect  differing  from  the  accounts  given  by 
professedly  formal  logicians. 

§  3.  Part  III  opens  new  ground.  Such  ontological 
conceptions  as  those  of  substance  and  causality — even 
of  'matter'  and  'mind' — are  explicitly  introduced  and 
their  significance  discussed  in  detail.  In  this  way,  a 
claim  is  made  that  logic  should  be  recognised  as  a 
department  of  philosophy  in  a  higher  sense  than  any 
warranted  by  the  restriction  of  its  scope  to  what  has 
been  termed  formal  logic.  It  is  true  that  inductive 
logicians  have  bestowed  much  care  upon  the  examina- 
tion of  the  nature  of  cause  and,  less  explicitly,  of  sub- 
stance. But  for  the  most  part  they  have  deliberately 
excluded  any  discussion  of  the  philosophical  implications 
attached  to  these  notions;  either  on  the  ground  that 
these  implications  belong  to  metaphysics  or  that  they 


4^ 


INTRODUCTION 


xvii 


4 


i 


4- 

4 


4 


are  to  be  rejected  in  toto  as  merely  bad  metaphysics. 
For  example,  though  much  of  what  Mill  has  said  and 
Venn  has  said  better  about  causal  and  other  uniformities 
has  its  value,  yet  it  is  obvious  that  their  treatment  gives 
us  no  instruction  on  the  philosophical  questions  at  issue. 
Moreover,  not  only  the  professedly  philosophical  logi- 
cians but,  strangely  enough,  also  the  humbler  inductive 
logicians  have  overlooked  or  devoted  insufficient  atten- 
tion to  many  methodological  problems  the  discussion 
of  which  belongs  to  the  logic  of  the  sciences.  This 
constitutes  my  apology  for  entering  with  considerable 
detail  into  topics  which  lie  on  the  borderland  between 
Logic  as  Methodology  and  Logic  as  Philosophy. 

The  inductive  logicians  may  be  said  to  have  presented 
a  philosophical  case  only  on  the  supposition  that  they 
are  to  be  interpreted  as  having  contended  for  the  inu- 
tility of  such  notions  as  those  of  causality  and  substance 
in  the  establishment  of  scientific  generalisations.  Thus 
Mill's  reduction  of  the  causal  relation  to  invariable  and 
unconditional  sequence  is  naturally  interpeted  as  tanta- 
mount to  the  rejection  of  the  notion  of  cause  in  any 
philosophical  sense.  And  this  is  certainly  the  contention 
of  those  among  later  empiricists  who  have  concerned 
themselves  with  the  problems  of  scientific  induction. 
In  fact,  the  more  modern  view  expressly  held  by  formal 
logicians  of  the  present  day  (who  are  mostly  empiricists 
of  the  school  of  Hume)  is  that  all  the  principles  of  in- 
duction (with  the  doubtful  exception  of  probability)  are 
derivable  by  an  extension  of  the  principles  of  deduction 
much  as  Pure  Mathematics  is  a  mere  extension  of  Pure 
Logic.  With  this  view  I  am  in  partial  agreement,  and 
the  discussions  of  Part  HI  are  largely  concerned  with 


xvm 


INTRODUCTION 


the  points  both  of  agreement  and  disagreement  between 
my  view  and  that  of  the  more  extreme  empiricists. 

In  examining  the  logical  foundations  of  science,  I  have 
found  it  impossible  to  separate  the  Epistemological  (or 
preferably  Epistemic)  from  the  Ontological  point  of  view. 
The  explanation  of  this  impossibility  is  that,  as  it  appears 
to  me,  certain  notions — and  certain  propositions  express- 
ible in  terms  of  these  notions — must  be  postulated,  if 
science  is  to  be  validly  established. 

By  a  postulate  I  understand  a  proposition  that  is 
assertorically  and  not  merely  hypothetically  entertained ; 
but  yet  is  adopted  neither  on  the  ground  of  intuitive 
self-evidence  nor  of  inductive  confirmation.  More 
positively,  a  postulate  is  framed  in  terms  not  given  in 
experience,  and  these  terms  enter  even  into  the  instan- 
tial  propositions  which  are  problematically  universalised 
by  induction.  Postulates,  in  my  view,  enter  even  into 
mere  observations  of  instances  which  may  otherwise  be 
termed  judgments  of  perception.  In  these  judgments 
the  thinker  predicates  not  merely  a  concomitance  of 
characters  presented  to  him ;  but,  besides  concomitance, 
causality;  and,  besides  presentment,  reference  to  sub- 
stance. 

§  4.  The  ontological  discussions  of  Part  III  are 
centred  upon  the  recognition  of  the  two  concepts, 
causality  and  substance.  But  I  have  discarded  the  term 
*  substance,'  for  reasons  which  need  no  enumeration,  in 
favour  of  the  term  'continuant'  The  genus  *  substantive 
proper,'  otherwise  termed  *  existent,'  is  divided  into  the 
two  species  'Continuant'  and  'Occurrent'  The  dis- 
tinction among  substantives  between  continuants  and 
occurrents  plays  a  similarly  prominent  part  in  material 


4i 


INTRODUCTION 


XIX 


i 


>4 


4 


i^ 


Y 


logic  as  is  played  in  formal  logic  by  the  distinction 
among  adjectives  between  determinables  and  determi- 
nates. But  no  analogy  can  be  drawn  between  the  anti- 
thesis or  connection  in  the  one  case  and  that  in  the 
other.  Negatively,  it  may  be  said  that  a  continuant  is* 
not  a  mere  collection  of  occurrentsjust  as  a  determinable  \ 
is  not  a  mere  collection  of  determinates.  Further  than 
this  we  can  only  say  that  a  plurality  of  occurrents  is 
constructed  by  thought  into  a  unity  by  virtue  of  the  nexus 
of  causality  and  a  plurality  of  determinates  by  virtue 
of  the  relation  of  opponency  or  incompatibility.  No 
positive  analogy  can  be  drawn,  owing  (it  would  seem) 
to  the  ultimately  irresolvable  antithesis  between  sub- 
stantive and  adjective. 

§  5.  A  more  detailed  summary  of  the  views  pro- 
pounded in  Part  III  on  ontological  problems  may  now 
be  given. 

In  the  first  place,  I  have  adopted  the  dualistic 
position  which  recognises  a  fundamental  distinction  be- 
tween the  psychical  and  the  physical,  and  attributes 
reality  to  both  in  the  same  unequivocal  sense.  Whether 
or  not  the  view  is  philosophically  tenable,  at  any  rate 
any  examination  into  the  principles  of  science  would 
seem  to  be  impossible  without  some  such  hypothesis  as 
that  of  dualism.  Spinoza's  acceptance  of  two  unsyn- 
thesised  attributes, — thought  and  extension — illustrates, 
in  more  or  less  veiled  guise,  the  very  same  fundamental 
position  as  that  adopted  by  the  dualist.  But  the  view 
that  I  wish  to  put  forward  is  less  dualistic  than  Spinoza  s, 
in  that  I  profess  to  present  the  psychical  and  the  phy- 
sical in  some  sort  of  synthesis  with  one  another,  and 
not  in  mere  unreconciled  antithesis.    What  I  hold  to 


** 


I 

I 


XX 


INTRODUCTION 


be  important  in  the  dualistic  position  is  the  recognition 
of  two  kinds  of  agency — psychical  agency  and  physical 
agency.  Of  my  views,  on  this  and  kindred  matters,  I 
do  not  profess  to  be  able  to  offer  any  direct  demonstra- 
tion, nor  do  I  believe  that  my  philosophical  opponents 
can  offer  any  valid  refutation.  The  more  detailed  ex- 
position of  my  philosophy  must  be  allowed  to  be  taken 
as  a  substitute  for  strict  demonstration. 

A  continuant  is  defined  to  be  that  which  continues  to 
exist  throughout  some  limited  or  unlimited  period  of 
time,  during  which  its  inner  states  or  its  outer  con- 
nections with  other  continuants  may  be  altering  or  may 
be  continuing  unaltered.  In  the  first  place,  then,  the 
continuant  must  be  contrasted  with  its  states — the  pos- 
sessive pronoun  here  pointing  to  a  unique  species  of 
*tie'  indicated  by  the  preposition  of  to  be  understood 
in  a  specific  sense  differing  from  all  other  senses.  There 
is  no  relational  word  (as  far  as  I  know)  that  can  be  used 
to  express  this  specific  meaning  of  *of,'  parallel  to  the 
relational  word  characterising  which  expresses  the 
specific  meaning  of  'of '  in  such  a  phrase  as  "the  quality 
^this  or  that."  In  fact,  the  two  meanings  of  the  word 
are  continually  combined  in  constructions  such  as 
those  expressed  by  the  phrase  **the  quality  of  this 
or  that  state  of  this  or  that  continuant."  Just  as  a 
quality  must  be  attached  or  referred  to  this  or  that 
state,  so  a  state  must  be  attached  or  referred  to  this 
or  that  continuant.  We  may  also  speak  of  a  property 
of  this  or  that  continuant  to  mean  a  property  char- 
acterising this  or  that  continuant,  so  that  property  (in 
this  application)  is  a  species  of  the  genus  adjective. 

Now  while  we  cannot  say  that  a  continuant  occurs, 


% 


INTRODUCTION 


XXI 


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we  can  say  that  a  state  occurs ;  and  anything  that  may 
be  said  to  occur  will  be  called  an  *  occurrent.'  And  I 
lay  it  down  that  any  occurrent  must  be  referred  to  a  con- 
tinuant or  to  two  or  more  connected  continuants.  The 
reference  of  an  occurrent  to  connected  continuants  will 
be  entailed  when  we  speak  of  transeunt  causality ;  while 
the  reference  of  an  occurrent  to  a  single  continuant  will 
be  entailecl  sometimes  in  speaking  oi  immanent  causality 
and  sometimes  in  speaking  of  transeunt  causality. 

§  6.    In  many  applications   *  occurrent'  and   'event 
may  be  taken  as  synonyms;  but,  properly  speaking 
they  must  be   distinguished.     Thus  what  is  called  a 
single  event  is  (or  may  be)  resolvable  into  a  plurality 
of  occurrents  of  different  kinds.    The  resolution  of  an 
event  into  a  plurality  of  occurrents  must  not  be  con 
founded  with  the  partition  of  an  event  into  a  pluralit 
of  parts.    The  parts  of  an  event  are  themselves  events'; 
and  these  are  distinguished  from  one  another  by  their 
difference  of  spatio-temporal  location.     On  the  other 
hand,  the  occurrents  composing  an  event   cannot  be 
distinguished  by  difference  of  location,  for  they  must  be 
located  within  the  same  spatio-temporal  boundaries  as 
the  event  itself. 

The  above  general  account  of  the  distinction  between 
occurrents  and  events  may  be  considered  first  in  regard 
to  physical  and  next  in  regard  to  psychical  events.  A 
physical  event  has  a  spatio-temporal  extension  which 
is  defined  by  the  spatio-temporal  boundary  within  which 
it  falls,  which  again  determines  the  four-dimensional 
magnitude  of  the  extension.  In  order  to  distinguish 
between  one  and  another  physical  event  it  would  seem, 
therefore,  both  necessary  and  sufficient  that  we  should 
J  L  ni  b 


xxti 


INTRODUCTION 


be  able  to  assign  different  spatio-temporal  boundaries 
to  the  two.  This  holds  even  of  the  event-parts  of  a 
whole  event  as  distinguished  from  one  another  and  from 
the  whole;  the  different  event-parts  being  said  to 
occupy  different  parts  of  the  extension  occupied  by  the 
whole  event.  Now,  besides  mentally  dividing  an  event 
into  parts,  we  may  also  mentally  resolve  an  event  into 
occurrents.  The  several  occurrents  which  thus  compose 
an  event  are  distinguished,  not  by  the  spatio-temporal 
position  which  they  occupy,  but  by  the  different  ad- 
jectival determinables  under  which  their  determinate 
characters  fall.  Now  all  that  is  here  said  about  physical 
events  and  physical  occurrents  holds  also  of  psychical 
events  and  psychical  occurrents,  except  for  the  fact  that 
spatial  reference  cannot  be  applied  to  the  latter  and 
temporal  reference  only  remains.  It  follows  that  the 
extension  of  a  psychical  event  and  the  magnitude  of  its 
extension  are  one-dimensional  instead  of  four-dimen- 
sional. Hence,  whereas  difference  of  position  would 
seem  to  be  necessary  and  sufficient  to  mark  off  one 
physical  event  from  another,  difference  of  dating  is  not 
necessary  or  sufficient  for  marking  off  one  psychical 
event  from  another.  Thus,  if  one  person  is  suffering 
tooth-ache  contemporaneously  with  another  person's 
reflecting  upon  a  mathematical  problem,  we  should 
speak  of  these  as  two  events,  although  we  cannot  attri- 
bute to  either  of  them  spatial  extension  or  boundary 
and,  therefore,  cannot  attribute  to  them  different  spatial 
extensions  or  boundaries. 

This  shows  that  in  order  mentally  to  separate  one 
psychical  event  from  another  we  must  postulate,  not 
only  a  difference  of  temporal  position  (if  any),  but  also 


< 


INTRODUCTION 


XXIU 


> 


■A 


> 


^ 


\ 


different  psychical  continuants  to  which  the  two  different  , 
psychical  events  are  to  be  referred.  A  priori,  indeed, 
the  same  must  hold  as  regards  physical  events;  i.e.  two 
simultaneous  events  might  occupy  the  same  localityi 
which  is  tantamount  to  the  possibility  that  two  bodies 
(physical  continuants)  should  be  *  occupying'  the  same 
place  at  the  same  time.  This  postulate  would  be  neces- 
sitated if  we  found  that  two  phenomena,  not  in  immediate 
causal  relation,  such  as  pressure  and  attraction  were 
occurring  at  the  same  place  and  at  the  same  time ;  just 
as  we  are  necessitated  to  postulate  two  psychical  con- 
tinuants when  two  psychical  events,  not  in  immediate 
causal  relation,  occur  within  the  same  period  of  time. 

§  7.  In  transeunt  causality,  as  so  far  expounded,  we 
conceive  two  continuants — which  in  the  first  instance 
are  to  be  physical — in  causal  connection  with  one 
another;  in  such  wise  that  the  alterable  *  state'  of  the 
one  continuant  is  attributed  as  effect  of  its  alterable 
relation  with  the  other.  This  conception  of  transitive 
causality  gives  significance  to  the  antithesis  'agent- 
patient.'  That  continuant  whose  'state'  is  occasioned 
by  its  relation  with  the  other  continuant  is  termed  (in 
this  connection)  patient,  and  that  continuant  whose 
relation  to  the  former  occasions  the  state  is  termed 
agent.  Logicians  who  have  rejected  the  antithesis  be- 
tween agent  and  patient  have  done  so  on  the  ground  1 
that  every  agent  is  at  the  same  time  patient  and  every 
patient  is  at  the  same  time  agent.  But,  even,  if  this  were 
universally  the  case,  the  distinction  would  remain;  since 
the  state  of  the  one  continuant  is  effect  of  its  relation 
with  the  other  continuant  while  the  concurrent  state  of 
the  other  continuant  is  effect  of  its  relation  with  the 

b2 


f 


XXIV 


INTRODUCTION 


INTRODUCTION 


XXV 


I 


former.  We  can  always  distinguish  between  the  one 
cause  which  occasions  its  effect  and  the  other  cause 
which  occasions  its  effect.  Hence,  I  should  substitute 
for  Kant's  three  categories  of  relation :  Continuant  and 
State;  Cause  and  Effect;  Agent  and  Patient. 

Several  points  in  the  consideration  of  transeunt  and 
immanent  causality  must  be  noted. 

(a)  Processes  which  are  immanent  to  a  whole  system 
of  interacting  continuants  may  always  be  regarded  as 
entailing  transeunt  causality  between  the  parts  of  the 
whole  system.  This  aspect  of  causality  is  familiar  to 
the  student  of  Physical  Science.  Or — to  express  the  same 
principle  in  converse  form — if  we  primarily  conceive  of 
interaction  between  parts  of  a  system  as  exhibiting 
transeunt  causality,  we  may  (without  contradiction)  ex- 
press our  formulae  in  terms  of  causality  immanent  to 
the  whole.  Physics  is  at  first  provisionally  monadistic, 
but  it  becomes  increasingly  monistic,  in  the  sense  that 
I  the  entire  range  of  physical  phenomena  come  to  be 
j  systematised  as  immanent  to  the  whole.  This  reduction 
!  of  the  whole  of  physical  reality  to  a  self-contained  system 
by  no  means  precludes  the  exposition  of  details  in  terms 
of  transeunt  causality. 

(h)  Now,  although  a  monistic  form  may  be  given  to 
the  system  of  all  physical  reality,  psychical  reality 
remains  essentially  pluralistic,  and  cannot  be  formulated 
monistically.  In  a  certain  sense,  physical  reality  exhibits 
the  kind  of  causality  that  is  termed  transeunt  and  no 
physical  causality  is  strictly  immanent.  This  is  because 
the  ultimate  constituents  of  matter — if  there  are  ultimate 
constituents — have,  so  to  speak,  no  insides.  A  physical 
event  must  always  and  can  only  be  described  as  a 


1 


i 


V 


)4 
k 


-iA 


1 


r 

'4 


i 


% 
A 


changing  or  unchanging  spatial  relation  of  one  thing  to 
another, — the  ultimate  *  thing'  having  no  inner  'states  j 
which  can  be  said  to  change  or  to  remain  unchanged." 
Hence,  the  immanency  ascribed  to  the  processes  occur- 
ring within  a  mentally  isolated  material  'body,'  is  only 
immanency  relative  to  processes  occurring  within  other 
mentally  isolated  material  'bodies.*  Nevertheless  the 
conception  of  immanency  cannot  be  eliminated  in  the 
formulation  of  physical  laws;  because  the  effects  upon 
one  body  due  to  transeunt  action  from  another  are  modi- 
fications of  what  would  be  happening  within  the  body 
were  no  such  transeunt  causality  in  operation.  Hence, 
the  analysis  of  transeunt  process  always  entails  reference 
to  immanent  process;  yet  the  converse  (as  it  seems) 
does  not  universally  hold ;  that  is  to  say,  it  seems  that 
purely  immanent  processes  occur  within  the  experiences 
of  a  single  Experient  (Psychical  Continuant),  though 
perhaps  never  within  the  happenings  of  a  single  Occu- 
pant (Physical  Continuant). 

§  8.  The  more  general  problem  in  regard  to  transeunt 
and  immanent  causality  relates  to  the  modes  in  which 
the  two  forms  operate  in  conjunction  with  one  another. 
When  any  complete  event  is  described  in  terms  both 
of  transeunt  and  of  immanent  causality,  it  would  appear 
that,  in  transeunt  causation,  the  cause-event  and  the 
effect-event  are  simultaneous;  but  that,  in  immanent 
causation,  the  cause-event  always  precedes  the  effect- 
event.  This  view  is  in  direct  contradiction  to  the 
prevailing  view  amongst  philosophers  who  profess  to 
attach  scientific  significance  to  the  antithesis  between 
the  transeunt  and  the  immanent.  Illustrations  in  support 
of  my  contention  will  be  found  in  the  body  of  my  work, 


XXVI 


INTRODUCTION 


i 


INTRODUCTION 


XXVll 


where  the  temporal  relations  between  cause  and  effect 
are  discussed.  Where  cause  precedes  effect,  as  in  im- 
manent causality,  I  hold,  in  agreement  with  other 
philosophers,  that  there  is  no  temporal  gap  between  the 
two;  they  are  strictly  contiguous  or  as  Dr  Broad  ex- 
presses it  adjoined.  Similarly,  in  transeunt  causality, 
so  far  as  spatial-relations  between  the  two  concerned 
continuants  can  be  assigned,  strict  spatial  contiguity 
goes  along  with  temporal  co-incidence.  The  above 
account  must  be  understood  to  be  preliminary  and  in  a 
sense  provisional;  for,  on  further  investigation,  it  will 
be  seen  that  the  simple  principle  that  I  have  laid  down 
must  be  partially  modified. 

§  9.  The  views  advanced  in  Part  1 1 1  on  the  problem 
of  mutual  interaction  between  *mind'  and  'body'  may 
here  be  sketched  in  outline;  and  it  should  be  said  at 
once  that  I  adopt  the  common-sense  dualistic  position 
and  am,  therefore,  largely  concerned  with  reconciling 
this  position  with  the  claim  of  science  to  have  succeeded 
in  formulating  psychical  and  physical  processes  in 
general  but  precise  terms.  The  common-sense  view 
expressed  briefly  is  as  follows.  Certain  physical  pro- 
cesses occur  in  accordance  with  purely  physical  laws 
and  are  unaffected  by  *mind*;  and  similarly  certain 
psychical  processes  occur  in  accordance  with  purely 
psychical  laws  and  are  unaffected  by  'body.'  Again, 
there  are  critical  instants  when  a  physical  cause  occasions 
a  psychical  effect  which  I  shall  term  a  sensation ;  and 
there  are  critical  instants  when  a  psychical  cause  which 
I  shall  term  a  volition  occasions  a  physical  effect.  Of 
these  last  two  cases,  the  former  I  shall  refer  to  under 
the  heading physico-psychical  causality;  the  latter, under 


I 


1 


A'l 


4) 

4 


V 


4' 


K 


r 

I 


4. 


K 
i 


the  heading  psychico-physical  causality.  Since  sensa- 
tions (immediately  occasioned  by  a  physical  cause)  often 
engender  psychical  processes  terminating  in  an  act  of 
volition  which  in  its  turn  initiates  a  physical  process; 
and  since  this  latter  sooner  or  later  produces  a  physical 
consequent  which,  at  a  critical  instant,  occasions  a  sen- 
sation, the  whole  system  of  action  and  interaction 
assumes  a  cyclic  form.  In  such  cases,  action  initiated 
from  either  side  is  followed  by  reaction  initiated  from 
the  other.  But  there  is  no  reason  to  suppose  that  the 
cycle  is  in  all  cases  completed.  On  the  contrary,  some 
stimuli  which  initiate  modification  of  sensation  are  not 
followed  by  a  consequent  volition  which  initiates  modi- 
fication in  the  physical  world ;  and  some  volitions  which 
initiate  modification  in  the  physical  world  are  not  followed 
by  a  consequent  stimulus  which  initiates  modification  of 
sensation.  Action  followed  by  reaction  is  probably  the 
exception  rather  than  the  rule. 

The  cyclic  processes  may  be  roughly  schematised  as 
exhibiting,  alternately,  transeunt  and  immanent  process. 
The  Greek  letters  ^  and\/i  indicate  respectively  'physical* 
and  'psychical'  occurrences,  and  an  arrow  stands  for 
'causing*  as  also  for  'preceding.'    Thus: 

^  (0       <^a ->  V'l -*  V's -^  <^6  > 

(2)       V^a -><^i -♦  <^2 ->  V't  • 

Here  the  action  (f)a-^^i  is  followed  by  the  reaction 
\lf^-><f)i,y  and  the  action  t//„-^^i  is  followed  by  the  reac- 
tion (f)2'-^\lji.  While,  moreover,  these  actions  and  re- 
actions illustrate/r^«^^^«/ causality,  the  intermediate  pro-i 
cesses  xjf^  -^  ^jj^  and  <^i  -><^2  I  shall  speak  of  as  immanent  i 
In  case  (2),  the  relation  of  the  originative  volition 
xff^  to  the  terminal  sensation  i/i^  illustrates  'purpose.' 


XXVIU 


INTRODUCTION 


INTRODUCTION 


XXIX 


f/ 


In  case  (i),  the  relation  of  the  physical  occurrence  (^^ 
(which  initiates  the  cycle)  to  the  physical  occurrence  (f}^ 
(which  terminates  the  cycle)  raises  a  general  problem 
which  is  as  yet  without  any  unanimously  accepted  solu- 
tion. This  problem  must  be  approached  from  a  new  side. 
The  problem  next  immediately  before  us  is  that  of 
psycho-physiological  parallelism.  The  term  'parallelism' 
is  the  well-known  figurative  equivalent  for  one-one 
correspondence  or  one-one  correlation.  But,  unfortun- 
ately, it  is  used  with  further  implications  of  meaning, 
two  of  which  are  in  flat  contradiction  with  one  another. 
In  philosophical  usage,  parallelism  is  generally  under- 
stood to  deny  causal  relation  between  the  psychical  and 
physiological  correspondents;  but,  in  Science,  no  such 
denial  is  implied  (except  of  course  by  those  scientists 
I  who  reject  causality  altogether  and  substitute  invari- 
ability). Now  the  grounds  for  maintaining  parallelism 
in  the  philosophical  sense  have  nothing  whatever  in 
common  with  those  for  maintaining  parallelism  in  the 
scientific  sense.  In  fact,  at  least  as  regards  neural  and 
sensational  processes,  most  uninstructed  persons  accept 
scientific  parallelism  and  would  (if  it  occurred  to  them) 
deny  philosophical  parallelism.  They  would  say  that, 
inasmuch  as  variations  in  sensation  correspond  to 
variations  in  neurosis  (as  they  are  informed  by  com- 
petent scientists)  the  former  variations  are  certainly 
caused  by  the  latter. 

§  lo.  Here  it  is  to  be  noted  that  the  scientific  assertion 
of  correspondence  is  one-sided,  whenever  (as  seems  in- 
evitable) the  notion  of  causality  is  superimposed  upon 
that  of  invariability.  Impartial  correspondence  would 
assert  that,  just  as  the  causal  antecedents  of  a  sense- 


% 


^A 


f 


•: 


stimulus — which  occasions  a  modification  of  sense- 
experience — are  purely  physical,  so  the  causal  antece- 
dents of  a  volition— which  occasions  a  modification 
in  the  physical  world — are  purely  psychical.  Scientists, 
however,  mostly  appear  to  maintain  that  it  is  a 
mere  illusion  to  suppose  that  the  processes  of  desire 
or  feeling  and  cognition  or  thought  which  terminate 
in  a  volition  are  causally  operative.  They  maintain 
that  the  really  operative  causality  resides  in  the  neural 
process  which,  in  accordance  with  the  correspon- 
dence theory,  accompanies  the  conative  and  cognitive 
experiences.  In  short,  whenever  the  psychical  processes 
V'l*  ^2»  ^3>  •••  follow  one  another  in  a  temporal  and 
invariable  order,  this  is  so  because  the  physical  pro- 
cesses <^i,  <^2»  ^3J  •••  follow  one  another  in  a  temporal  and 
invariable  order.  They,  thus,  tacitly  maintain  a  one- 
sided operation  of  transeunt  causality.  They  assert 
that  the  sequence  <^i -><^2 ~* <^3  constitutes  the  cause  of 
the  sequence  ^i  -^  ^2  -*  V's*  ^^^  ^his  assertion  entails  that 
the  sequence  ^1  ->  1/^2  -^  ^3  never  constitutes  the  cause  of 
the  sequence  <^i— x^a— x^a-  Adapting  our  previous 
schematisation  to  the  present  problem,  the  scientists' 
view  would  be  indicated  thus: 

^1  ->  V'i 
T      t 


in  contrast  with 


^i-*^2 


i       i 

where  the  vertical  arrows   (in  both  cases)   stand  for 
transeunt  causality. 


XXX 


INTRODUCTION 


INTRODUCTION 


XXXI 


Of  course,  if  causality  were  excluded  altogether,  so 
that  the  vertical  arrows  stood  merely  for  simultaneity 
and  the  horizontal  arrows  merely  for  sequence,  then 
there  would  be  no  relevant  distinction  between  the  two 
alternative  modes  of  representing  the  facts.  Now,  the 
view  of  alternate  action  and  reaction  is  partially 
expressed  by  saying  that,  in  some  cases  <^i  and  ^2  ^^' 
spectively  cause  xjj^  and  xfj^y  while  in  other  cases  xjj^  and  xjj^ 
respectively  cause  (f)^  and  <^2-  That  is  to  say  in  cases 
where  xfj^,  xjj^y  etc.  stands  for  a  sequence  of  sensations 
then  these  are  related  to  the  sequence  of  neural  processes 
<^i,  <^2»^tc.  as  effect  to  cause.  But  in  cases  where  i/ij,  1/^2  >^tc. 
stand  for  a  course  of  conative  and  cognitive  deliberation, 
then  (if  this  course  is  accompanied  by  any  discoverable 
physiological  processes  corresponding  to  the  course  of 
the  psychical  processes)  t/ij ,  ^^,  etc.  are  related  to  <^i,  ^^ 
as  cause  to  effect. 

In  Part  Ilia  still  bolder  view  is  put  forward:  viz. 
that  just  as  there  are  countless  cases  in  which  physical 
processes  do  not  immediately  occasion  any  psychical 
processes  whatever,  so  there  are  cases  in  which  psychical 
processes  do  not  immediately  occasion  any  physical 
I  process  whatever.  This  view  may  be  termed  impartial 
i  dualism.  Or — expressing  the  same  view  in  metaphorical 
but  familiar  language — what  is  maintained  is  that  man  is 
a  genuinely  causal  agent  in  reference  to  which  his 
bodily  organism  serves  directly  and  materials  outside 
his  organism  indirectly  as  instruments  of  his  will  On 
this  view,  a  volition  is  immanently  caused  by  such 
purely  psychical  processes  as  feeling,  desire,  knowledge 
and  thought  to  which  there  are  no  neural  or  physiological 
correspondents. 


^V 


I' 

V 


^ 

r 

# 

M 
.1 


f 


§  1 1.  Before  attemptingtogivedirectevidence  in  sup- 
port of  the  theory  of  impartial  dualism,  the  scientific  objec- 
tions to  this  view  must  first  be  met.  Physical  Science 
claims  that,  in  such  a  cycle  as  (^^  — ►  i//i  ->  1//2 . . .  ->  V/„  ->  ^^ 
theoretically  completed  knowledge  would  be  able,  from 
the  physical  nature  of  <^a>  to  infer  the  physical  nature  of 
<^5,  apart  from  any  reference  to  the  intermediate  psychical 
occurrences  xjj^,  xjj^,  ...  V'n-  The  chain  of  events  would 
assume  the  form  (f>a-^<f>i-^<l>2~^^n-^4^bf  where  ^1, 
<^2 , . . .  <^^  would  represent  assignable  physiological  pro- 
cesses occurring  within  a  given  bodily  organism.  Now 
the  impartial  dualist  may  fullyadmit  this  contention  of  the 
physical  scientist  and  yet  adhere  to  the  view  which  attri- 
butes genuine  causality  to  the  'mind.'  For,  the  initial  cause 
<f>ay  which  operates  from  without  the  particular  organism, 
does  not  enable  science  to  infer  the  terminal  effect  c^^, 
without  consideration  of  the  special  sequence  <^i,  <^2,... 
(^^  which  varies  according  to  the  special  nature  of  the 
organism.  The  form  of  response  or  reaction  set  up  in 
one  organism  (expressed by  <^i,<^2»  •••  <^n)  differs  from  that 
set  up  in  another.  These  differences  must  be  taken  into 
consideration  if  the  specific  nature  of  the  effect  ^^  is  to 
be  inferred.  We  must  causally  account  for  the  differences 
in  the  intra-organic  processes  as  between  one  organism 
and  another.  This  account  will  entail  reference  to  the 
past  history  of  the  individual  organism  and  of  its  an- 
cestors. But  what  is  the  nature  of  the  cause  that  stamps 
upon  this  or  that  organism  its  own  special  mode  of 
organic  response?  This  speciality  of  response  can  be 
predicted,  by  means  of  ascertained  rules  of  uniformity 
framed  in  purely  physical  terms;  but  why  are  such  or 
such  physical  antecedents  invariably  followed  by  such 


i 


XXXll 


INTRODUCTION 


INTRODUCTION 


XXXIU 


-V 


or  such  physical  consequents  ?  The  character  stamped 
upon  each  organism — by  reference  to  which  alone 
physical  effects  can  be  inferred  from  physical  causes — 
may  be  the  consequent  of  psychical  processes,  operating 
in  such  invariable  modes  as  (theoretically  at  least)  can 
be  formulated  in  terms  of  physiological  habits  or  trends 
or  properties.  The  supremacy  of  physical  law  within  the 
whole  range  of  the  physical  is  not  hereby  overthrown 
when  mind  is  taken  to  be  a  genuinely  efficient  agent ; 
for  the  notion  of  law  may  imply  mere  invariability, 
whereas  that  of  an  agent  implies  causality. 

§  12.  A  consideration  of  the  different  ways  in  which 
invariability  and  causality  may  be  logically  related  gives 
rise  to  some  questions  of  the  greatest  philosophical  im- 
portance. In  some  cases,  we  have  well-assured  ground 
for  asserting  invariability,  and  from  such  assurance 
venture  precariously  to  infer  causality.  In  other  cases, 
we  have  well-assured  ground  for  asserting  causality,  and 
from  such  assurance  venture  precariously  to  infer 
invariability.  The  former  type  of  case  is  that  in  which 
our  main  reliance  is  upon  the  accumulation  of  wide  and 
varied  instantial  evidence ;  the  latter,  that  in  which  our 
main  reliance  is  upon  the  precision  and  accuracy  with 
which  we  can  analyse  single  instances.  The  distinction 
between  these  two  types  of  logical  procedure  is,  I  be- 
lieve, roughly  illustrated  in  many  regions  of  scientific 
enquiry.  But  I  wish  to  maintain  that  this  logical  dis- 
tinction can  be  applied  as  a  ground  of  division  between 
two  departments  of  knowledge.  By  direct  introspection, 
I  feel  assured  that  I  can  assign  the  cause  of  any  one  of 
my  acts  of  will ;  but  it  is  only  with  considerable  doubt 
that  I  should  venture  to  formulate  rules  in  accordance 


f' 


I 

Vj 

t 


J. 


{ 


with  which  I  invariably  act.  In  virtue  of  this  assurance 
I  maintain  that,  in  willing,  I  am  both  free  and  deter- 
mined :  determined,  because  my  volition  is  not  uncaused ;) 
free,  because  the  immediate  causal  determinants  of  my 
volition  are  within  my  own  consciousness. 

Causal  determination  of  the  will  cannot  be  based  on 
the  ground  of  any  observable  uniformity  of  behaviour  on 
the  part  of  myself  or  of  mankind  in  general  or  of  animals. 
This  is  partly  because  no  universally  applicable  rules  of 
behaviour  can  be  formulated ;  but,  more  obviously,  be- 
cause I  do  not  know  in  what  precise  points  the  deter- 
mining antecedents  of  one  action  agree  with  or  differ 
from  those  of  another.  In  order  to  formulate  rules  of 
behaviour  or  conduct,  I  must  obtain  accumulative  evi- 
dence upon  which  a  precarious  generalisation  may  be 
inductively  grounded ;  and,  when  all  that  is  conceivably 
possible  has  been  carried  out  by  inductive  procedure, 
my  reliance  rests  ultimately  upon  the  direct  assurance 
of  causal  determinism  yielded  by  introspection. 

§  13.  The  above  analysis  is  open  to  the  charge  of 
extreme  naiveU,  But,  before  attacking  my  position  on 
this  or  other  grounds,  I  ask  my  readers  to  note  that  my 
account  of  the  will  differs  in  some  important  respects 
from  those  given  by  others.  Many  disputants  on  the 
subject  of  freedom  of  the  will  have  put  determinism  and 
freedom  in  antithesis,  whereas  the  true  antithesis  is  be- 
tween determinism  and  indeterminism.  This  latter 
antithesis  was  (I  believe)  first  explicitly  put  forward  by 
Dr  G.  Ward,  who  was  still  more  explicitly  followed  by 
Pearse  and  W.  James.  Sidgwick  declares  that  in 
immediate  consciousness  we  are  assured  of  freedom, 
but  he  goes  on  to  maintain  that  the  determinism  that 


'M^ 


XXXIV 


INTRODUCTION 


INTRODUCTION 


XXXV 


seems  to  be  almost  demonstrated  by  a  sort  of  induction 
contradicts  the  freedom  that  is  introspectively  revealed. 
Again,  many  writers  who  reject  determinism,  interpret 
determinism  as  being  materialistic: — a  view  which  I 
absolutely  disclaim.  Again  Mill  and  others  reject 
freedom  on  the  ground  that  it  assumes  the  effects  of 
volition  to  be  known  a  priori  without  recourse  to 
experience ;  whereas  the  freedom  which  I  maintain  en- 
tails rather  direct  knowledge  of  the  immediate  causes 
of  volition.  The  knowledge  of  which  I  have  direct 
assurance  is  a  knowledge  of  the  purely  psychical  phases 
such  as  desire  and  cognition  of  which  I  can  become  aware 
by  retrospective  or  introspective  attention ;  and  these 
factors  present  themselves  to  me  as  cause  of  this  or  that 
volition.  I  am  quite  ignorant  of  the  physiological  pro- 
cesses which  issue  in  an  overt  physical  movement ;  and  it 
is  only  after  actual  experience  that  I  can  foresee  the  more 
or  less  remote  physical  effects  of  any  act  of  will,  as  is 
abundantly  established  by  psychological  enquiry.  And 
again  it  is  only  by  means  of  an  extended  experience 
that  I  can  venture  to  generalise  with  respect  to  the 
volitions  which  will  follow  upon  any  recurrence  of  the 
same  externally  presented  conditions,  since  the  intensity 
of  my  desires  and  the  determinateness  of  my  cognitions 
are  subject  to  alterations  in  the  course  of  time. 

One  other  frequent  misrepresentation  of  the  question 
under  dispute  must  be  mentioned.  It  is  alleged  against 
the  determinist  that  he  has  falsely  attributed  to  the  will 
a  kind  of  causality  which  is  borrowed  from  the  mechani- 
cal type  of  causation  appropriate  only  to  physical 
phenomena;  whereas,  in  truth,  as  history  proves,  it  is  the 
type  of  causation  exhibited   in  human  volitions  that 


1\ 


I 


L 


\ 


) 


c 


has   been   borrowed   and  falsely  applied   to  physical 
phenomena. 

§  14.  Some  justification  is  needed  for  my  devoting 
so  large  a  space  to  the  detailed  discussion  of  such 
psychological  or  metaphysical  topics  as  freedom  and 
determinism  in  a  work  professedly  logical.  My  excuse 
is  that  the  psychological,  metaphysical  and  logical  aspects 
of  these  problems  have  not  been  properly  disentangled ; 
and  that  it  is  only  by  bringing  these  aspects  into  close 
connection  with  one  another  that  we  shall  succeed  in 
getting  to  the  root  of  the  matter.  Many  empirical 
psychologists  have  explicitly  put  forward  the  view  that, 
whether  or  not  freedom,  in  some  metaphysical  sense,  is 
to  be  attributed  to  the  will,  at  any  rate  psychologists 
must  work  on  the  hypothesis  of  determinism.  In  this 
way,  they  preclude  any  discussion  as  to  whether 
psychological  determinism  is  or  is  not  incompatible 
with  metaphysical  freedom.  Or  again :  Kantians  have 
tried  to  reconcile  transcendental  freedom  with  empirical 
determinism.  But  this  attempt  needs  a  preliminary 
discussion  of  the  logical  relation  between  freedom  and 
determinism  ;  and,  moreover,  attributes  freedom  to  the 
transcendental  ego  and  determinism  to  the  empirical 
ego.  Now,  in  a  philosophical  treatment  of  such  scienti- 
fic conceptions  as  those  of  substance  and  causality, 
there  is  no  place  for  a  transcendental  ego  or  any  species 
of  Ding  an  sick.  The  freedom  attributed  by  science  to 
the  will  is  empirical  in  just  the  same  sense  as  that  in 
which  determinism  is  attributed.  What  causally  deter- 
mines any  act  of  volition  is  a  temporal  event  or  process 
manifesting  the  character  of  the  psychical  agent,  just  as 
what  causally  determines  a  physical  consequent  is  a 


I « 


XXXVl 


INTRODUCTION 


temporal  event  or  process  manifesting  the  character  of 
physical  agents. 

In  order,  then,  to  present  a  consistent  and  compre- 
hensive view  of  the  philosophical  principles  underlying 
scientific  constructions  and  inferences,  it  is  necessary  to 
examine  in  what  way  such  conceptions  as  cause  and 
substance  and  such  antitheses  astranseuntand  immanent 
causality  are  actually  employed  in  science.  The  form 
in  which  these  conceptions  enter  into  psychical  science 
fundamentally  agrees  with  and  also  fundamentally  differs 
from  that  in  which  they  enter  into  physical  science. 
Problems  of  parallelism  and  interaction  could  not  be 
fruitfully  discussed — even  in  a  preliminary  logicar  sur- 
vey— without  entering  into  controversial  detail  when 
attempting  to  apply  the  logical  points  at  issue  to  the 
scientific  analysis  of  psychical  and  physical  facts. 


1 


'.^ 


I  / 


j*» 


4.     . 


,u 


CHAPTER  I 


FACT  AND  LAW 


§  I .  Assertions  about  the  universe  of  reality  fall  into 
two  distinct  classes  which  may  be  designated  ( i )  asser- 
tions of  fact  and  (2)  assertions  of  law : — where  the  terms 
fact  and  law  are  restricted  to  the  sense  in  which,  taken 
together,  they  include  experientially  certifiable  propo- 
sitions and  exclude  formal  propositions.  Other  terms 
approximately  synonymous  to  *fact'  and  'law'  are  'con- 
crete' and  'abstract,'  or  again  'categorical'  and  'hypo- 
thetical'; but  these  terms  are  used  too  loosely  to  bring 
out  the  antithesis  which  rests  really  upon  the  funda- 
mental distinction  and  relation  between  substantive  and 
adjective.  Although  according  to  our  analysis  every 
proposition  is  to  be  interpreted  in  terms  of  both  sub- 
stantive and  adjective,  we  may  assert  provisionally  that 
in  the  abstract  proposition  or  assertion  of  law,  the  ad- 
jective is  the  more  explicit  or  solely  explicit  factor, 
whereas,  in  the  concrete  proposition  or  assertion  of 
fact,  the  substantive  is  the  more  explicit  factor.  Asser- 
tions of  fact  may  be  statements  either  of  a  single  fact, 
i.e.  about  a  single  substantive,  or  of  several  single  facts 
summarised  in  a  proposition  which  shall  have  the  same 
factual  nature  as  the  several  propositions  of  which  it  is 
a  summary.  Or  again,  a  concrete  proposition  may  ex- 
press not  a  conjunction  but  an  alternation  of  single 
facts,  and  in  this  case  it  will  be  of  the  same  nature  as 
the  assertions  that  constitute  the   several    alternants. 


J  L  ni 


g 


\ll 


2  CHAPTER  I 

though  less  determinate  than  any  one  of  them.  In  dis- 
cussing the  nature  of  a  factual  proposition  th|en,  we  need 
only  consider  the  proposition  which  expresses  a  single 
fact,  without  conjunction  or  alternation.  The  subject  term 
of  such  a  proposition,  which  denotes  a  pure  substantive 
without  adjectival  characterisation,  is  best  symbolised  as 
S,  and  *5  is/'  will  stand  for  a  single  assertion  of  fact 
where/  is  the  adjective  characterising  the  substantive  5. 
§  2.  The  first  difficulty  about  the  proposition  *5  is/ ' 
relates  to  what  we  may  call  the  referential  problem: 
in  other  words,  to  what  subject  is  the  predicate/  to  be 
referred  when  we  assert  '  5  is  /  '  ?  For,  if  the  symbol 
5  is  non-significant — and,  in  default  of  any  adjectival 
characterisation,  it  is  difficult  to  see  what  significance 
it  can  have — then  the  proposition  'S  is  /'  cannot  be 
intelligently  distinguished  from,  say,  the  proposition 
'  Tisp'  where  T^is  equally  non-significant  with  S.  If 
we  agree  that  '  5  is  / '  and  ^  T  is  / '  are  different  propo- 
sitions, we  may  yet  look  beyond  them  for  a  common 
class  to  which  both  terms  5  and  T  belong.  This  com- 
mon class  is  denoted  by  the  wide  term  substantive  used 
in  its  very  general  sense ;  hence,  as  a  further  interpre- 
tation of  our  formulae,  the  two  propositions  to  be 
distinguished  may  be  rendered  in  the  forms  '  This  sub- 
stantive is  /'  and  '  That  substantive  is  /.'  The  intro- 
duction of  the  terms  'this'  and  'that'  serves  to  show 
that  substantives  can  be  distinguished  apart  from,  and 
independenriy  of,  any  adjectival  characterisation;  so 
that,  starting  with  'this  substantive'  and  'that  sub- 
stantive '  we  may  complete  our  predication  by  asserting  ^ 
of  'this'  or  of  'that'  either  the  same  or  a  different 
adjective.    As  I  have  stated  elsewhere,  I  regard  the 


I 


r^ 


V 


I 


> 


\ 


FACT  AND  LAW  3 

principle  of  distinction  which  is  independent  of  cha- 
racterisation as  ultimatelyTbased  on  the  psychological 
fact  of  separateness  of  presentment  of  the  manifesta- 
tions of  reality.  The  predesignation  'a  certain'  best 
indicates  this  separateness  of  presentment;  and  thus 
the  more  adequate  formulation  of  the  factual  proposi- 
tion runs:  'A  certain  given  manifestation  is  /.'  The 
introductory  indefinite  being  preparatory  to  the  refer- 
ential definite,  we  pass  from  the  predesignation  *a  cer- 
tain' to  the  definite  'this'  or  'that.'  This  transition  is 
possible  psychologically  so  far  as  we  can  identify  and 
discriminate  the  positions,  temporal  or  spatial,  at  which 
manifestations  are  presented  in  separateness ;  and  such 
identification  or  discrimination  of  position  is,  I  maintain, 
psychologically  prior  to  any  subsequent  relating  in  space 
or  time,  no  less  than  to  all  forms  of  qualitative  charac- 
terisation. The  significance  of  the  word  'given'  in  our 
formula  is  two-fold;  in  the  first  place,  it  indicates  all 
that  is  meant  by  the  word  'real';  and  in  the  second 
place,  it  anticipates  the  general  nature  of  the  charac- 
terisation which  completes  the  predication.  For  what 
is  given,  otherwise  called  the  determinandum,  is  pre- 
sented under  a  certain  determinable,  symbolisable  by 
the  capital  letter  P  corresponding  to  the  little  letter/. 
The  process  of  thought  being  the  further  determination 
of  the  relatively  indeterminate,  a  further  amendment  of 
the  formula  will  be:  'A  certain  given  /*  is  /.'  Those 
logicians  who  wish  to  introduce  identity  into  their  an- 
alysis of  the  proposition  may  be  partially  gratified  by 
this  recurrence  of  the  same  letter  in  both  subject  and 
predicate^;  but  the  fact  that,  ultimately,  the  subject  term 

1  See  Part  II,  Chapter  I,  §  9. 

I — 2 


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\ 


t 


\ 


4  CHAPTER  I 

represents  indeterminately  what  is  represented  deter- 
minately  in  the  predicate  term,  does  not  preclude  the 
referential  problem  of  the  singular  categorical  propo- 
sition; a  problem  which  has  been  met  by  the  unique 
employment  of  the  phrase  'a  certain'  which  is  prepara- 
tory to  the  definite  'this*  or  'that/  So  much  for  the 
factual  proposition. 

§  3.  Passing  to  the  consideration  of  the  abstract  pro- 
position or  assertion  of  law,  this  may  be  expressed 
purely  in  terms  of  characterising  adjectives,  in  the  form 
'p  determines  q!  Here  the  word  'determine'  demands 
special  consideration.  In  our  account  of  the  simple 
categorical  statement  of  fact,  we  spoke  of  determina- 
tion by  thought,  and  to  apply  determination  in  this 
sense  to  our  abstract  proposition,  we  should  have  to 
combine  the  abstract  assertion  '/  determines  q'  with 
the  concrete  assertion  'a  certain  P  is/,'  these  two  pro- 
positions together  determining \is  to  assert  'xhisPisq.' 
According  to  this  interpretation  of  the  word  'determine,' 
the  abstract  proposition  may  be  said  to  express  an 
anticipatory  determination  for  thought ;  for  it  must  be 
conjoined  with  the  concrete  proposition  in  order  to  de- 
termine any  further  assertion. 

Many  logicians  have  been  satisfied  with  this  merely 
epistemic  account  of  the  relation  of  determination — a 
position  which  is  tantamount  to  identifying  the  thought 
relation  implication,  with  the  causal  relation  in  its  widest 
sense,  as  indicated  by  Hume's  phrase  'objective  nexus.' 
Here  we  may  note  that  Kant,  deliberately  opposing 
Hume,  took  the  relation  of  implication  to  apply  only  to 
thought  in  general,  and  to  be  the  typical  form  of  judg- 
ment corresponding  to  the  category  of  causality,  the 


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4 


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FACT  AND  LAW  5 

causal  relation  having  validity  in  an  objective  sense.  In 
this  contention  Kant  undoubtedly  aimed  at  distinguish- 
ing the  subjective  or  epistemic  from  the  objective  or 
constitutive  relation;  but  on  this  matter  of  the  very 
first  importance  his  view  has  been  very  variously  in- 
terpreted. Of  all  the  interpretations  I  shall  adopt  that 
in  which  the  two  conceptions  of  determination  are 
most  widely  opposed.  Before  entering  into  the  detailed 
analysis  of  this  position,  we  must  refer  back  to  the 
epistemic  distinction  between  experiential  and  formal 
certification.  For  example,  an  arithmetical  formula, 
expressing  relations  between  numerical  adjectives,  is 
one  that  can  be  formally  certified  apart  from  particular 
experiences.  In  contrast  to  this,  any  proposition  which 
formulates  a  law  of  nature  can  only  be  certified  ulti- 
mately by  means  of  particular  experiences.  Now  in 
Mill's  use  of  the  phrase  'empirical  uniformity'  there 
seems  to  me  to  be  involved  a  fundamental  confusion 
between  the  epistemic  and  the  constitutive  points  of 
view  which  it  is  immediately  necessary  to  remove. 
Epistemically  understood  Mill's  phrase  points  to  the 
ultimate  data,  namely  observed  instances,  upon  which 
the  generalisation  under  consideration  is  based;  and 
since  he  holds  that  all  generalisations  about  natural 
phenomena  are  established  on  this  same  basis,  there 
should  be  no  distinction  for  him  between  empirical  uni- 
formities and  causal  laws.  Mill  nevertheless  hints  at  an 
ontological  distinction  between  these  two  kinds  of  uni- 
formity where,  for  instance,  he  asserts  that  the  method  j 
of  agreement  cannot  prove  causal  laws ;  for  if,  as  seems 
probable,  in  using  this  phrase  he  meant  the  emphasis  to 
fall  on  the  words  'causal  law,'  he  must  have  had  an 


iC 


7 


i 


r/ 


r  •-: 


! 


6  CHAPTER  I 

ontological  distinction  in  mind;  it  is  only  if  the  empha- 
sis were  upon  the  word  'prove'  that  a  purely  epistemic 
point  arises.  The  same  confusion  is  apparent  in  his 
view  that  the  causal  relation  involves  not  only  invaria- 
bility but  unconditionality.  In  my  own  view  this  quali- 
fication of  Mill's  represents  the  ontological  distinction 
between  a  universal  of  fact  and  a  universal  of  law. 
Thus  taking  the  two  determinate  adjectives  p  and  q 
under  the  respective  determinates  P  and  Q,  the  factual 
universal  may  be  expressed  in  the  form  *  Every  sub- 
stantive PQ  in  the  universe  of  reality  is  q\if\  while 
the  assertion  of  law  assumes  the  form  *  Any  substantive 
PQ  in  the  universe  of  reality  would  be  q  if  it  were  // 
These  formulae  represent  fairly,  I  think,  the  distinction 
which  Mill  had  in  mind;  for  my  first  formula  may  be 
said  to  express  a  mere  invariability  in  the  association 
of  q  with  /,  while  the  second  expresses  the  uncondi- 
tional connection  between  q  and  p.  Or,  as  I  have  said 
in  p.  252,  Chapter  xiv.  Part  I,  the  universal  of  fact 
covers  only  the  actual,  whereas  the  universal  of  law  ex- 
tends beyond  the  actual  into  the  range  of  the  possible. 

§  4.  Now  the  introduction  of  the  word  'possible' 
here  requires  us  to  summarise  briefly  the  main  senses 
in  which  the  word  is  used  in  common  thought  and  in 
philosophy : 

{a)  The  possible  may  be  understood  as  equivalent 
to  what  is  capable  of  being  construed  in  thought;  in 
this  sense  it  is  equivalent  to  the  conceivable.  Now  the 
effort  to  construe  in  thought  an  entity  which  has  been 
expressed  in  verbally  intelligible  form  can  be  analysed 
into  a  step  by  step  process  such  that  the  combination 
of  characters  and  relations  constructed  up  to  a  certain 


4i 


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if 

J     ■* 


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FACT  AND  LAW  7 

point  may  present  to  us  some  further  character  which 
our  thought  is  compelled  to  assign  to  the  construction. 
What  then  constitutes  the  impossibility  of  the  proposed 
construction  is  the  attempt  to  replace  this  further  cha- 
racter, which  we  were  compelled  to  predicate,  by  another 
character  which  is  positively  opponent  to  the  former.   It 
is  this  positive  opponency  between  characters,  there- 
fore, which  constitutes  the  genuine  inconceivability  upon 
which  non-existence  is  to  be  maintained.  In  other  words, 
the  impossibility  of  some  one  mental  construction  is  • 
derivative  from  the  necessity  of  a  contrary  or  opponent  I 
mental   construction.    Let  us   take  the  most  familiar 
example:  the  non-existence  of  a  collection  defined  at 
the  same  time  as  two  plus  three  and  as  seven,  does  not 
depend  direcdy  upon  the  impossibility  of  mentally  con- 
joining these  two  numerical  predications,  but  indirecriy 
upon  the  necessity  of  conjoining  the  predication  two 
plus  three  with  the  predication  yfz^^,  of  which  seven  is  a 
positive  opponent  or  contrary.     It  is  not   a  question 
of  difficulty— amounting  to  an  apparent  impossibility— 
of  making  a  thought  construction  in  accordance  with  a 
verbal    formula   that   constitutes   inconceivability  and 
gives  the  true  test  of  non-reality ;  but  rather  the  posi-l 
tive  necessity  of  making  some  determinate  construction) 
opponent  to  the  proposed  construction. 

{b)  A  second  meaning  of  the  word  possible  is  quite 
easy  to  define;  it  relates  merely  to  the  limitations  of 
knowledge :  so  that  we  say  it  is  possible  that  such  or 
such  may  be  the  case,  meaning  to  express  the  quite 
simple  fact  that  we  are  not,  at  the  time,  able  to  make 
a  positive  assertion  concerning  the  truth  or  falsity  of 
the  proposed  proposition.    In  this  sense  of  the  word 


8 


CHAPTER  I 


^ 


possible,  there  is  nothing  in  the  nature  of  the  proposi- 
tion itself,  apart  from  person  and  circumstance,  which 
can  determine  its  being  possibly  true  or  not,  and  for  it 
I  prefer  to  substitute  the  word  problematic.  A  special 
case  of  this  type  of  possibility  arises  when  an  indi- 
vidual has  in  his  possession  knowledge  of  various  truths 
which  he  has  not  combined  in  thought,  so  as  to  elicit  by 
mere  thought  process  some  further  truth.  In  default  of 
this  thought  process,  the  proposition  expressing  this 
further  truth  is  not  known,  and  is  therefore  possibly 
true  and  possibly  false  for  him.  All  the  complicated 
formulae  of  mathematics  and  logic  come  within  this 
class  for  the  ordinary  man  who  has  not  taken  occasion, 
or  who  is  intellectually  incapable,  of  developing  such 
knowledge.  This  consideration  leads  to  a  third  meaning 
of  possibility. 

(c)  Propositions  may  be  said  to  be  possibly  true  or 
possibly  false,  in  an  explicitly  referential  sense;  that  is 
to  say,  possibility  here  is  a  feature  not  intrinsic  to  the 
proposition  itself,  but  only  when  considered  in  reference 
to  some  other  body  of  propositions  taken  to  be  true. 
Any  proposition,  then,  whose  falsity  or  truth  cannot  be 

i  formally  deduced  from  a  given  body  of  propositions, 
may  be  said  to  be  possibly  true  and  possibly  false 
referentially  to  this  body. 

(d)  The  further  meanings  of  the  word  possible  are 
connected  with  the  notion  of  natural  law  and  its  anti- 
thesis to  what  we  have  called  fact.  The  general  form 
of  a  law,  exhibiting  the  constitution  of  nature,  has  been 
expressed  'U  any  substantive  were  characterised  as/ 
it  would  be  characterised  as  ^.'  This  proposition  ex- 
presses a  relation  between  the  characters  p  and  ^  indi- 


hi 


^ 


4k 


If 


F 
0 


i 


FACT  AND  LAW  9 

cative  of  the  nature  of  the  world  of  reality.  If  any  two 
characters  ;t;and^  are  no^  so  related,  then  the  conjunction 
of  X  with  any  opponent  of  y  would  be  said  to  be  a 
possible  conjunction.  When  speaking  of  any  fact  or 
'  event  as  distinguished  merely  by  spatio-temporal  position 
from  other  facts  or  events,  such  terms  as  necessary  or 
contingent  cannot  be  applied.  On  the  other  hand,  when 
we  describe  the  event  by  an  enumeration  of  certain  ad- 
jectives or  characteristics  finite  in  number,  and  there- 
fore non-exhaustive,  the  nomic  distinction  between  the 
necessary  and  the  contingent  has  significance  relatively 
to  such  description  of  the  fact,  though  not  relatively  to 
the  fact.  Thus  the  fact  may  be  described  as  2.fqr  which 
is  X,  And  so  described  it  will  be  nomically  necessary 
provided  that  any  substantive  characterised  by  pqr 
would  be  characterised  by  x ;  but  it  would  be  nomically 
contingent  if  anything  characterised  by  pqr  were  not 
necessarily  x.  Now  the  nomic  necessity — anything 
characterised  by  pqr  would  be  characterised  by  x — 
implies  the  factual  universal  that  '  everything  that  is 
pqr  is  actually  x' \  whereas  the  nomic  contingency 
'  anything  that  is  pqr  is  not  necessarily  x,'  does  not 
imply  the  factual  particular  that  *  some  things  that  are 
pqr  are  not  x' \  i.e.  the  affirmation  of  law,  or  nomic 
necessity, implies  the  factual  universal;  but  the  negation 
of  law — i.e.  the  affirmation  of  nomic  contingency — does 
not  imply  the  factual  particular.  The  logicians  who 
reject  the  contrast  that  I  maintain  between  law  and 
fact,  identify  in  effect  nomic  necessity  with  the  universal 
of  fact,  and  nomic  contingency  with  the  particular  of  fact. 
The  conflict  between  these  two  views  is  apparent  in  the 
special  case  in  which  a  factual  universal  expresses  only 


10 


CHAPTER  I 


a  contingency;  that  is  to  say,  when  *  Every /^r  is  x' 
goes  along  with  *  Any  pqr  might  be  not-^ ' :  e.g.  the 
merely  factual  universal  that  *  Every  day  is  followed  by 
night '  is  compatible  with  the  statement  of  contingency 
that  *Any  day  might  be  not  followed  by  night.'  Now% 
the  possibility  of  joining  these  two  statements  depends 
upon  day  being  defined  by  a  definitely  limited  con- 
junction of  characters;  for,  if  our  definition  exhausted 
all  the  characters,  it  would  render  the  sequence  of  night 
inevitable,  and  we  should  be  confronted  with  a  universal 
of  law.  Expressing  this  symbolically: — An  event  de- 
scribed merely  as  a  pqr  that  is  x  may  represent  a 
contingency j^jthough  such  an  event  ^ouldjtheoretically 
always  be  more  fully  described  as  a  pqruvw  which  is 
necessarily  x.  It  may  appear,  since  by  an  adequate 
description  a  contingency  thus  becomes  a  necessity, 
that  the  notion  of  nomic  contingency  has  therefore  no 
application.  But,  if  we  consider  precisely  why  the  con- 
ditions uvw,  say,  have  to  be  added  to  the  conditions /^r, 
in  6rder  that  ;trmay  necessarily  follow,  it  is  because /^r 
does  not  nomically  necessitate  uvw^  and  therefore  that 
the  relation  of  pqr  to  uvw  is  nomically  contingent. 
Thus  the  abstract  question  whether  the  character  x  of 
the  given  event  is  necessary  or  not  is  unanswerable, 
since  it  is  seen  to  be  contingent  relatively  to  the 
incomplete  description  pqr ;  and^necessary  relatively 
to  the  complete  description  pqruvw.  The  philosophical 
justification  of  the  principle  under  consideration  requires 
the  postulate  that  any  character  such  as  x  manifested 
in  a  particular  event  is  ontologically  dependent  upon 
an  assignable — and  therefore  finitely  enumerable — set 
of  characters  pqruvw. 


i 

4.. 


^> 


I 

I 


I 


r 


V 


> 


4 
i 


f. 


FACT  AND  LAW 


IX 


§  5.    In  further  explication  of  the  formal  distinction 
between  the  assertion  of  law  and  the  ordinary  or  factual 
universal,  we  require  to  define  the  class  expressed  by 
the  phrase  *  Anything  that   might  be  /';    for  there 
are  limitations  to  this  class.    Thus,  if  there  is  a  law 
of  nature  that  anything  that  may  be  x  would  neces- 
sarily be  not-/,  then  a  thing  defined  as   in  the  class 
X  would  be  excluded  from  the  class  of  things  that  might 
be/,  and  this  class  includes,  not  anything  whatever,  but 
only  such  things  as  have  been  defined  by  a  character 
not  necessarily  precluding  /.     Now  a  class  defined  in 
this  way  is  very  different  from  an  ordinary  or  factual 
class ;  for  we  cannot  take  a  given  case  and  say  whether 
it  belongs  to  the  class  or  not ;  all  we  know  of  a  thing 
whose  character  is  determined  as  x,  say  (where  x  corre- 
sponds to   '  Anything  that  might  be  / '),  is  that  this 
character  x  would  not,  under  the  realm  of  natural  law, 
prohibit  its  being  /.    The  particular  case  in  question 
might,  however,  have  other  characteristics  which  wou/d 
prohibit  its  being/:  thus,  if  it  be  characterised  as  x 
and  jv,    where  the  character  x  does  not  prohibit  /, 
while  the  character  y  does  prohibit  its  being  /,  and 
we  take  the  completed  definition  xy  of  the  class  to 
which  the  thing  belongs,  it  could  not  possibly  be  /  ; 
but  if  we  take  the  incomplete  definition,  which  includes 
only  X  and  drops  y,  then  we  may  assert  of  the  thing 
that  it  might  possibly  be  /.    A  concrete  illustration 
will  make  this  point  clear  :— Let  x  stand  for  a  railway 
journey,  and  let  us  suppose  further  that  in  any  actual 
railway  journey  the  train  travels  with  a  brake.   Now  as 
far  as  the  definition  railway  journey  is  concerned,  there 
is  nothing  that  prevents  the  train  travelling  without  a 


> 


12 


CHAPTER  I 


brake,  and  therefore  any  instance  might  be  one  of  a 
brakeless  train.    If  we  now  take  jj/  to  stand  for  the  pre- 
caution which  actually  prevents  the  train  being  brake- 
less,    the  complete  determination  of  the  case  would 
preclude  it  from  being  one  of  a  brakeless  train.    The 
proposition  to  which  our  illustration  leads  may  be  put 
in  the  form  :  '  Any  railway  journey  with  a  brakeless 
train  would  be  liable  to  accident,'  and  the  force  of  this 
proposition  obviously  extends  over  what  must  be  called 
a  wider  range  than  the  whole  class  of  actual  railway 
journeys ;   if  we  assume  as  a  matter  of  fact  that  all 
railway  journeys  use  a  brake.  This  illustration  suggests 
a  wide  class  of  cases  which  indicate  human  foresight  or 
prudence  ;  and  in  all  such  cases  the  distinction  between 
the   nomically  necessary  and  the  factual  universal  is 
quite  apparent.  If  any  action  that  might  be  characterised 
as   such  or  such    would   produce   undesirable  conse- 
quences, and  if  human  conduct  is  actually  determined 
by  knowledge  of  such  consequences,  then,  as  a  matter 
of  fact,  those  actions  will  never  take  place.  The  nature 
of  the  actual  occurrences  is  defined,  on  the  one  hand, 
by  the  circumstances  which  would  make  such  or  such 
conduct  disastrous,  and  on  the  other  hand  by  the  know- 
ledge on  the  part  of  mankind,  of  this  fact.     If  the  oc- 
currence  be  defined  only  by  the  circumstances,  we  can 
say  of  it  that  it  mz^-A^  be  such  or  such ;  but  if,   to 
complete  the  determination  of  the  case,  we  add  the 
knowledge  of  the  consequences  on  the  part  of  mankind, 
then  this  complete  determination  prohibits  the  possi- 
bility of  its  being  characterised  as  such  or  such. 

An  example  resembling  that  of  the  brakeless  train, 
is  *Any  person  caught  trespassing  on  this  field  will  be 


I 


FACT  AND  LAW 


13 


i^ 


^■rm 


u 

A 

^ 

^ 


44 


! 


P  i, 


\ 


f. 


{ 


prosecuted.'  This  proposition  applies  not  only  to  the 
persons  who  have  actually  been  caught  trespassing  and 
who  have  therefore  been  prosecuted,  but  to  persons 
who  have  trespassed  and  have  not  been  caught;  for  it 
is  true  of  these  latter  persons,  as  much  as  of  the  former, 
that  if  they  had  been  caught  they  would  have  been 
prosecuted.  The  application  of  the  proposition,  there- 
fore, again  extends  to  the  possible,  and  is  not  restricted 
to  the  actually  existent ;  though  this  illustration  differs 
from  the  other  inasmuch  as  there  are  cases  of  uncaught 
trespassers,  whereas  we  supposed  that  no  train  travelled 
without  a  brake. 

Another  illustration  of  the  same  principle  may  be 
taken  from  the  sphere  of  physics.    Thus  from  such  a 
formula  as — *  retardation  varies  as  the  coefficient  of 
friction ' — it  follows  that  if  the  coefficient  of  friction 
were  reduced  to  zero,  the  retardation  would  be  zero. 
But  in  actual  fact  there  is  no  instance  in  which  the 
movement  of  one  body  over  another  does  not  entail 
friction,  so  that  the  above  proposition  applies  over  a 
range  beyond  actual  fact.    The  point  of  importance, 
therefore,   is  that  an  assertion  of  this  type  may  be 
scientifically  established  as  true,  while  there  may  be  no 
case  presented  in  fact  to  which  it  is  actually  applicable. 
If  propositions  of  this  kind  were  interpreted  as  merely 
existential  or  factual,  the  actual  non-existence  of  the  class 
defined  by  the  subject  term  would  render  it  a  matter  of 
indifference  whether  one  or  any  other  predicate  term 
were  substituted.    It  follows  that  a  merely  factual  or 
existential  interpretation  of  this  type  of  proposition  is 
totally  inadequate,  and  that  to  express  its  significance 
the  proposition  must  be  understood  as  applicable  to  the 


14 


CHAPTER  I 


wide  range  of  what  is  possible,  as  contrasted  with  the 
narrower  range  of  what  is  actual. 

§  6.  There  is  a  subtle  case  in  which  the  notion  of  the 
epistemically  possible  and  the  contingent — i.e.  nomically 
possible — are  combined,  indicated  by  the  term  potential 
in  one  of  its  applications.  Thus  using  the  symbols 
employed  above,  we  may  know  of  a  particular  object 
that  it  is  pqr,  and  that  being  such  it  may  also  be  x ;  and 
further  that  if  it  were  uvw  as  well  as  pqr  it  would 
necessarily  be  :r.  In  this  sense  we  may  say  that  its 
being  x  is  hypothetically  necessary — a  term  which 
Mr  Bradley  uses  to  define  the  possible.  Here  the  force 
of  the  word  hypothetical  is  purely  epistemic,  and  as 
thus  applied  it  means  that  we  do  not  know  whether 
the  thing  is  or  will  be  uvw,  knowing  only  that  it  is  pqr. 
The  term  necessary,  however,  is  used  ontologically  or 
nomically,  and  means  that  anything  that  is  pqruvw 
would  necessarily  be  x.  Now  I  have  to  maintain  that, 
given  pqr,  xcsinnot  be  said  to  be  hypothetically  necessary 
unless  it  is  possible  that  anything  that  is  pqr  may  also 
be  uvw.  We  cannot  therefore  define  the  possible  as 
equivalent  to  the  hypothetically  necessary,  because  the 
proposed  character  x  is  not  even  hypothetically  neces- 
sary unless  the  junction  of  pqr  with  uvw  is  itself 
nomically  possible.  The  meaning  of  the  term  potential, 
then,  when  the  given  thing,  known  to  he  pqr,  is  said  to 
be  potentially  x,  involves  first  epistemic  possibility, 
i.e.  we  must  not  know  that  it  t's  uvw;  and  secondly 
nomic  possibility,  i.e.  anything  that  is  pqr  may  be 
uvw.  The  most  important  use  of  the  term  potential 
coming  under  this  wide  head  requires  reference  to  a 
continuant  subjected  to  transeunt  causality.    Thus  to 


I 


FACT  AND  LAW 


15 


y 


V- 


r 


i" 


\ 


\ 


t- 


r 

■i 


say  of  a  solid  strong  body  that  it  is  potentially  capable 
of  resisting  a  certain  measurable  degree  of  pressure 
implies,  from  the  epistemic  point  of  view,  that  our 
known  data  do  not  include  the  knowledge  that  such  a 
big  pressure  will  be  actually  applied  ;  that  such  a  force 
may  be  applied  is  therefore  hypothetical,  or  more  pre- 
cisely, epistemically  possible.  But  further,  to  convey 
the  full  significance  of  potential  resistance,  this  epistemic 
possibility  must  be  combined  with  the  negative  fact 
that  there  is  nothing  in  the  laws  of  nature,  and  in 
particular  in  the  character  of  the  body  itself,  which 
would  prevent  this  large  force  being  applied.  Shortly, 
then,  the  potential  resistance  of  a  body  means  epistemi- 
cally that  we  do  not  know  whether  a  certain  force  will 
be  applied  or  not ;  and  ontologically  or  nomically,  that 
there  is  nothing  in  the  nature  of  things  to  prohibit  such 
force  being  applied. 


4 


} 


^.•" 


CHAPTER  II 

THE  CRITERIA  OF  PROBLEMATIC  INDUCTION 

§  I.  That  induction  is  the  inverse  of  deduction  is  a 
truism ;  but  it  is  worth  while  to  develop  this  truism  in 
its  various  aspects.  In  its  simplest  form  this  inverse 
relation  is  exhibited  by  the  change  of  place  of  premisses 
and  conclusion,  for  roughly  deductive  inference  consists 
in  the  passage  from  /AH  P's  are  ^ '  to  '  Certain  given 
P's  are  Q ' ;  and  inductive  inference  in  the  passage  from 
Certain  given  P's  are  Q'  to  All  Ps  are  Q,'  Whereas 
this  deductive  inference  is  formally  demonstrative,  the 
inductive  inference  is  obviously  only  problematic,  and 
in  general  of  a  low  degree  of  probability.  The  logical 
theory  of  induction  may  be  developed  by  showing  in 
what  respects  the  degree  of  probability  of  such  an  in- 
ductive conclusion  depends  on  the  aggregate  nature  of 
the  instances  examined. 

The  general  procedure  in  an  inductive  process  is  as 
follows :  certain  given  instances  are  noticed  as  being 
characterised  by  certain  adjectives — say  P  and  Q — and 
we  proceed  to  look  for  other  instances  characterised  by 
P,  in  order  to  discover  whether  they  are  also  charac- 
terised by  ^.  In  the  preliminary  stages  of  induction, 
where  P  and  Q  jointly  characterise  certain  observed 
instances,  the  sole  factor  which  decides  us  to  search  for 
other  instances  of  P  in  order  to  discover  whether  they 
are  Q,  rather  than  for  other  instances  of  Q  to  discover 
whether  they  are  P,  is  that  we  have  observed  instances 


s 


kM 


I 


r 

<     4 


> 


i 


\ 
w 


THE  CRITERIA  OF  PROBLEMATIC  INDUCTION      17 

of  Q  which  are  not  P,  while  so  far  every  observed  in- 
stance of  P  has  been  Q.  In  other  words,  we  already 
know  that  '  Not  all  Q's  are  /*,'  and  therefore  our  enquiry 
is  restricted  to  the  question  whether  'All  Ps  are  QJ 
The  search  for  new  instances  to  which  we  are  thus 
prompted  constitutes  the  preliminary  process  called 
discovery,  and  these  instances  are  presented  to  us  in 
one  or  other  of  two  ways.  They  may  either  occur  in 
the  course  of  nature,  and  be  discovered  by  active  search 
in  appropriate  places  and  at  appropriate  times ;  or,  on 
the  other  hand,  we  may  have  the  means  of  producing 
them  at  places  and  in  times  where  the  course  of  nature, 
if  uninterfered  with,  would  not  have  exhibited  such 
instances.  These  two  kinds  of  active  search  are  briefly 
denominated  non-experimental  and  experimental :  both 
imply  activity  prompted  and  guided  by  a  definitive 
purpose. 

§  2.  The  use  of  experiment  in  discovery  can  only  be 
accounted  for  by  anticipating  a  discussion  of  what  is  to 
be  understood  by  the  somewhat  vague  term  'uniformity.' 
We  speak  of  experiment  as  an  interference  with  the 
course  of  nature ;  but  we  do  not  in  any  sense  conceive 
that  by  such  interference  the  laws  or  uniformities  of 
nature  are  violated  ;  for  of  the  laws  or  uniformities  of 
nature  we  may  provisionally  say  that  they  do  not  pre- 
scribe the  dates  and  places  at  which  phenomena  will 
occur,  except  so  far  as  the  dates  and  places  at  which 
other  phenomena  have  occurred  have  been  determined. 
\{  then  the  human  will  has  the  power  of  directly  pro- 
ducing phenomena,  the  course  of  nature  is  modified, 
though  the  uniformities  of  nature  are  not  infringed. 
The  phrase  'uniformity  of  Nature'  thus  involves  a 
J  L  ni  ^ 


jf  ■■* 


<• 


i8 


CHAPTER  II 


"11 


certain  ambiguity :  it  may  mean  uniformity  in  the  course 
of  nature  independently  of  man's  interference ;  as,  for^ 
example,  in  the  continued  elliptic  motion  of  the  planets,  • 
or  in  the  upward  convection  of  heat  through  the  atmo- 
sphere ;  or  it  may  mean  the  aggregate  of  the  laws  or 
uniformities  which  are  obeyed  even  when  there  has  been 
human  interference ;  as,  for  example,  in  the  construction 
and  working  of  Foucault's  pendulum,  or  in  carrying  a 
hot  body  from  one  place  to  another.  Whether,  as  in  the 
first  case,  there  is  no  interference,  or  as  in  the  second 
case,  human  purpose  intervenes,  the  laws  of  gravity  and 
of  convection  of  heat  are  equally  unviolated ;  and  in 
either  case,  the  phenomena  observed  will  afford  means 
for  studying  the  uniformities  in  accordance  with  which 
the  operations  of  nature  take  place. 

Further,  from  a  certain  point  of  view,  uniformity  in 
nature  holds  even  when  man  interferes  ;  for  nature  in- 
cludes man,  and  we  shall  here  assume  that  voluntary 
action  obeys  laws  which  as  such  are  psychological,  and 
exhibit  the  nature  of  man  himself.  Thus,  if  we  suppose 
the  occurrence  of  a  definite  purpose  to  be  the  immediate 
cause  of  the  time,  place,  and  manner  of  a  certain  inter- 
ference in  the  course  of  nature,  the  formation  of  this 
purpose  may  be  assumed  to  have  depended  upon  ante- 
cedent psychological  conditions,  and  thus  to  exhibit  the 
kind  of  uniformity  which  is  characteristic  of  man  in  his 
capacity  of  voluntary  agent.  The  use  of  man's  power 
to  interfere  with  the  course  of  physical  nature  is  prompted 
not  only  by  the  purpose  of  acquiring  further  knowledge, 
but  also  by  utilitarian  ends.  Thus  the  face  of  the  physical 
world  is,  at  the  present  time,  totally  different  from  what 
it  would  have  been  if  the  laws  of  physical  nature  alone 


\ 


^\ 


>    -*• 


r*. 


r 


THE  CRITERIA  OF  PROBLEMATIC  INDUCTION      19 

had  been  in  operation.  We  must  therefore  recognise  a 
partial  independence  together  with  a  mutual  interaction 
between  psychical  and  physical  process,  each  following 
its  own  laws  and  also  affecting  the  phenomena  of  the 
other.  And  this  interdependence  resolves  the  apparent 
paradox  of  scientific  experiment,  which  consists  in  inter-- 
fering  with  the  course  of  nature,  with  the  purpose  of 
discovering  more  determinately  the  laws  of  nature. 

§  3.  Returning  now  to  the  search  for  new  instances ; 
it  may  be  assumed,  in  the  simplest  case,  that  all  the  in- 
stances of/  so  far  examined  have  been  discovered  to 
be  q ;  or,  rather,  more  precisely,  that  none  of  them 
have  been  discovered  to  be  other  than  q ;  for  the 
character  q  may,  in  certain  instances  in  which  we  have 
detected  p,  be  beyond  our  power  of  observation.  We 
thus  arrive  at  an  enumerative  universal,  'All  examined 
/'s  are  q^  and  this  proposition  constitutes  the  inductive 
premiss  from  which  we  venture  to  infer  with  a  lower 
or  higher  degree  of  probability  that  'AH  p's  are  q' 
Now  this  summary  or  enumerative  premiss  may  have 
very  different  degrees  of  value  as  evidence  for  the 
universal ;  we  will  therefore  proceed  to  sketch  in 
outline  the  different  tests  by  which  its  value  may  be 
estimated. 

In  the  first  place,  if  this  positive  premiss  stood  alone 
it  would  in  general  have  very  little  value ;  only  when 
it  is  combined  with  one  or  more  complementary  pro- 
positions which  taken  together  mutually  support  one 
another,  has  it  serious  evidential  value.  To  constitute 
such  a  complementary,  a  proposition  must  have  as  its 
subject  term  a  substantive  with  characters  opposed  to 
that  of  the  subject  of  the  positive  premiss.    This  intro- 


/ 


\ 


20 


CHAPTER  II 


•V 


duces  the  general  notion  of  the  determinable  and  its 
opposed  determinates,  and  is  explained  in  detail  in  the 
chapter  on  eduction,  where  I  discuss  the  employment 
of  intensional  and  extensional  intermediaries.  Here  we 
will  simply  point  out  that,  in  calling  a  set  of  premisses 
complementary,  we  are  extending  or  modifying  the  use 
of  that  term  from  the  sense  in  which  it  denotes  pairs  of 
propositions  like  '  Every  p  is  q'  and  '  Every  non-p  is 
non-^,'  so  as  to  cover  an  indefinite  number  of  pro- 
positions corresponding  to  the  indefinite  number  of 
values  of  P  which  have  been  correlated  with  an  equal 
number  of  different  values  of  Q.  Thus  the  inductive 
premiss  may  be  represented  as  a  set  of  complementary 
enumeratives : 

Every  examined  p  '\s  q 
Every  examined/'  is  q' 
Every  examined  /"  is  q" 


and  corresponding  to  these  premisses,  the  inductive 
conclusion  may  be  represented  as  a  set  of  comple- 
mentary universals  : 

Every/  is  q 
Every  p'  is  q' 
Every  p"  is  q" 


It  must  not  be  supposed  that  each  several  of  these 
premisses  constitutes  by  itself  the  evidence  for  the 
corresponding  universal ;  on  the  contrary,  the  several 
premisses  taken  jointly  constitute  the  experiential  data 
upon  which  the  strength  of  evidence  for  each  of  the 
several  universals  depends.  This  kind  of  compound 
induction  then,  which  aims  at  discovering  evidence  that 


I 

\ 


"> 


I 


"  I 


** 


r 


THE  CRITERIA  OF  PROBLEMATIC  INDUCTION      21 

the  value  of  Q  depends  upon  the  value  of  P,  does  so 
by  finding  that  in  all  examined  cases  the  value  of  Q  has 
varied  along  with  variations  in  the  value  of  P^  and  that  I 
it  has  been  found  constant  whenever  P  was  constant. 
Thus  the  notion  of  dependence  has  two  sides:  (1)  that 
the  constancy  of  the  one  variable  entails  the  constancy 
of  the  other;  and  (2)  that  the  variation  of  the  one 
variable  entails  a  variation  of  the  other.  The  reader 
will  note  that  the  collection  of  data  in  compound  in- 
duction of  this  type  roughly  resembles  Bacon's  Table 
of  Degrees  and,  somewhat  less  closely,  Mill's  Joint 
Method  of  Agreement  and  Difference. 

§  4.  In  rough  or  pre-scientific  induction  of  the  kind 
just  described,  it  is  not  assumed  that  we  are  dealing  with 
simplex  variables,  nor  even  with  complex  variables  that 
have  been  analysed  into  their  simplex  factors :  thus  on 
further  analysis  we  might  afterwards  discover  that  (say) 
p  =  aUcy  that  /'  =  dUc,  and  that  /"  =  aU'd ;  and  similarly 
with  q.  This  leads  to  the  consideration  of  another  cri- 
terion which  affects  the  cogency  of  inductive  inferences, 
viz.,  what  will  be  called  the  criterion  of  specification. 
For  example:  in  the  inference  from  *  Every  examined 
p  is  q'  to  'Every/  is  q^  there  is  a  liability  to  generalise 
too  widely — a  danger  which  is  great  in  proportion  to 
the  indeterminateness  of  the  subject  character/;  hence 
the  more  specifically/  can  be  defined,  the  less  hazardous 
will  be  our  generalisation.  The  question  arises:  How 
specifically  the  determinate  character/  must  be  defined 
in  order  to  limit  this  generalisation  ?  Now  the  different 
cases  which  we  have  examined  will  all  have  agreed  in 
certain  characters,  while,  as  regards  other  characters, 
some  instances  will  have  differed  from  others.  We  may 


\ 


is 


22 


CHAPTER  II 


therefore  conceive  of  a  certain  conjunction  of  characters 
— S3.yabcd — which  characterise  every  examined  instance, 
and  by  including  all  these  characters  in  our  definition 
of  the  subject  term,  we  limit  our  generalisation  within 
strictly  logical  bounds.  In  the  case  before  us  the  con- 
clusion will  then  assume  the  form  *  Every  abed  is  q' ; 
and  this  strict  specification  prevents  us  from  inferring 
any  wider  generalisation  such  as  *  Every  abc  is  ^,'  or 
*  Every  bed  is  q,'  or  the  still  wider  generalisation  'Every 
be  is  q'  An  elementary  illustration  will  help  to  explain 
the  force  of  this  principle  of  specification.  Common 
experience  had  afforded  mankind  in  early  times  in- 
variable evidence  of  unsupported  bodies  falling  to  the 
earth ;  if  from  this  they  had  inferred  that  all  unsupported 
bodies  would  fall  to  the  earth,  they  would  have  neglected 
a  character  common  to  all  the  observed  instances — 
namely  that  of  proximity  to  the  earth ;  their  generalisa- 
tion ought  therefore  to  have  been  restricted  to  the 
statement  that  every  unsupported  body  in  proximity  to 
the  earth  would  fall.  There  are  many  phenomena  which 
can  be  observed  by  man  in  a  region  of  space  limited  in 
some  such  way  as  this  ;  and  hence  the  generalisations 
based  upon  such  observations  should  be  limited  to  the 
regions  in  which  the  character  is  manifested.  ^  Of  course 
this  does  not  mean  that  natural  phenomena  are  de- 
pendent upon  absolute  spatial  conditions,  but  only  that 
there  may  be  material  bodies,  occupying  particular 
regions  of  space,  upon  which  the  phenomena  depend. 
The  same  applies  to  periods  of  time :  absolute  dating 
in  time  does  not  affect  natural  phenomena ;  but  there 
may  be  types  of  events  occurring  within  certain  periods 
of  time,  upon  which  other  occurrences  within  those 


->wr     1^ 


r 


i' 


THE  CRITERIA  OF  PROBLEMATIC  INDUCTION     23 

periods  depend.  Thus,  under  the  general  principle  of 
evolution,  the  forms  in  which  the  uniformity  of  nature 
is  manifested  will  be  very  different  at  different  periods ; 
it  would  therefore  be  invalid  to  infer,  from  the  recorded 
evidence  that  nations  throughout  history  have  been 
either  preparing  for  or  actually  engaged  in  war,  that 
this  will  be  the  case  in  future :  not  because  absolute 
time,  any  more  than  absolute  space,  is  relevant  for  the 
uniformities  of  nature,  but  because  the  occurrences 
within  a  particular  period  of  time  causally  affect  other 
occurrences  within  that  period ;  just  as  material  bodies 
within  a  certain  region  of  space  causally  affect  other 
material  bodies  within  that  region. 

The  principle  of  specification  can  only  be  approxi- 
mately realised  in  practice ;  for  practically  it  demands 
that  the  instances  examined  shall  agree  with  one  another 
in  no  characters  over  and  above  those  which  are  used 
to  define  the  range  of  the  generalisation.  But  if  all  the 
instances  of  abed  for  example,  agree  in  only  one  or  two 
other  characters,  say  mv,  our  generalisation,  though  it 
ought  strictly  to  be  limited  to  the  narrower  class  abeduv, 
may  perhaps  be  safely  extended  to  the  whole  class  abed. 
The  generalisation  approximates  to  certainty  in  pro- 
portion as  the  additional  characters  common  to  the 
examined  instances  decrease  in  number.  Thus  the 
principle  of  specification,  expressed  in  familiar  language, 
demands  that  an  assortment  of  instances  designed  to 
establish  a  generalisation,  should  be  as  varied  as  possible 
within  the  range  defined  by  the  characters  comprised 
in  the  subject  term.  In  this  form  it  is  seen  to  be 
practically  equivalent  to  the  principle  underlying  the 
method  of  agreement,  which  requires  that  the  instances 


24 


CHAPTER  II 


upon  which  a  generalisation  is  based  should  exhibit 
together  the  maximum  of  difference  or  of  variety. 

§  5.  The  principle  of  specification  applies  to  a  single 
enumerative  such  as  *  Every  examined  abed  is  ^';  but 
we  have  seen  that  the  condition  for  the  highest  degree 
of  probability  is  that  the  generalisation  in  question 
should  be  supported  by  a  set  of  complementary  enu- 
meratives,  and  we  proceed  to  consider  what  relations 
should  subsist  between  these  several  enumeratives. 
The  requisite  condition  in  this  case  is  that,  under  the 
several  different  enumeratives,  the  instances  examined 
should  agree  with  one  another  as  closely  as  possible  in 
all  characters  other  than  those  in  which  they  are  known 
to  differ.  Thus  within  the  same  enumerative,  the  in- 
stances should  differ  from  one  another  as  far  as  possible 
in  the  other  characters;  and  under  different  enumera- 
tives, they  should  agree  with  one  another  as  far  as 
possible  in  the  other  characters.  This  latter  requisite 
approximates  in  practice  to  Mills  method  of  difference, 
application  of  which  demands  that  the  instances 
examined  should  agree  as  far  as  possible.  We  thus 
have  two  complementary  criteria,  the  one  requiring 
variety,  and  the  other  similarity.  These  principles 
express  the  common  practice  of  the  uninstructed  mind; 
we  can  only  justify  them  when  we  enter  into  the  rela- 
tion of  probability  to  induction. 

§  6.  So  far  we  have  not  referred  to  the  number  of 
instances  examined  as  a  criterion  affecting  the  strength  of 
evidence.  I  n  point  of  fact  mere  number  does  not  directly 
strengthen  the  instantial  evidence ;  its  importance  de- 
pends upon  variety;  and  number  counts  only  because,  by 
increasing  the  number  of  instances  under  the  method  of 


V 


>  f 


>  k- 


\ 


A  -^1 


*r 


\ 


f 


-*     ^ 


4 
i 


f 


t*^ 


THE  CRITERIA  OF  PROBLEMATIC  INDUCTION      25 

agreement,  their  variety  is  probably  thereby  increased.| 
Corresponding  to  number,  what  is  required  for  instances 
under  the  method  of  difference  may  be  denominated 
proximity ;  not  because  mere  proximity  in  time  or  space 
is  important,  but  because  instances  which  are  either 
temporally  or  spatially  close  to  one  another  will  probably 
agree  in  many  characters  which  it  may  be  impossible 
for  us  to  analyse.  Thus  the  analogous  criteria  of  number 
and  proximity  are  both  only  inferior  substitutes  for 
analysis.  When  under  agreement  we  cannot  analyse 
sufficiently  the  characters  of  the  instances  to  enable  us 
to  assert  difference  between  them  in  many  respects,  we 
have  to  rely  upon  mere  number  of  instances,  which  are 
presumed  to  secure  a  probable  maximum  of  difference. 
Similarly  under  the  method  of  difference,  when  the 
elaborate  analysis  required  to  enable  us  to  assert  agree- 
ment in  many  respects  between  the  instances  is  im- 
practicable, we  have  to  rely  upon  m^r^ proximity  which 
is  presumed  to  secure  a  probable  maximum  of  agree- 
ment. The  term  proximity  here  is  to  be  understood 
to  include  besides  what  would  be  literally  called  spatial 
or  temporal  proximity,  also  reference  to  the  same 
agent  whose  conditions  are  varied  from  instance  to 

instance. 

§  7.  The  fact  that  the  criteria  of  number  and  proximity 
are  mere  inferior  substitutes  for  the  more  direct  criteria 
of  variety  and  similarity,  at  once  suggests  that  the 
evidential  value  of  examined  instances  really  depends 
upon  the  extent  to  which  our  analysis  enables  us  to 
assert  agreement  or  difference  in  the  characters  of  the 
compared  instances  ;  and  the  probability  of  a  generalisa- 
tion therefore  varies  with  the  degree  of  precision  with 


.V 


26  CHAPTER  II 

which  we  are  able  to  define  the  characters  of  the  in- 
stances examined.  This  consideration  throws  H^ht  upon 
Mills  problem  'why,  in  some  cases,  a  single  instance 
is  sufficient  for  a  complete  induction,  while  in  others, 
myriads  of  concurring  instances,  without  a  single  ex- 
ception known  or  presumed,  go  such  a  very  little  way 
towards  establishing  a  universal  proposition/    Speaking 
in  terms  of  mere  number,  intensional  number  is  of  much 
higher  value  than  extensional  number ;  that  is  to  say 
the  number  of  characters  in  which  instances  are  known 
to  agree  and  differ  is  of  much  greater  evidential  im- 
portance than  the  actual  number  of  instances  examined. 
But  again  the  mere  number  of  characters  analysed  is 
not  directly  important  in  itself,  any  more  than  the  mere 
number  of  instances  examined :  the  characters  counted 
should  be  strictly  independent  of  one  another,  and  this 
requirement  is  exacdy  parallel  to  that  which  demands 
that  instances  examined  should  vary  with  one  another. 
Any  character  whose  presence  is  dependent  upon  the 
conjunction  of  a  given  set  of  characters  adds  nothing 
to  their  evidential  value.   And,  similarly,  any  instance 
which  agrees  with  a  given  set  of  instances  in  all  the 
respects  in  which  these  agree  with  one  another  adds 
nothing  to  their  evidential  value.    Therefore  whenever, 
in    order    to    construct   the   intensional   criterion   for 
problematic  induction,  we  count  characters  which  are 
not  known  to  be  independent,  we  are  relying  upon  the 
likelihood  that  a  good  many  of  them  are  independent. 
And  when,  to  constitute  the  extensional  criterion,  we 
merely  count  instances  which  are  not  known  to  be  per- 
tinendy  different,  we  are  relying  upon  the  likelihood 
that  a  good  many  of  them  are  pertinendy  different. 


/ 


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4' 


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A^ 


vr 


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Si 


I 


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-4. 


4    *i 


i 


t^*- 
^ii 


THE  CRITERIA  OF  PROBLEMATIC  INDUCTION      27 

§  8.  The  above  parallel  applies  to  instances  and  their 
characters  under  what  may  be  in  general  called  the 
method  of  agreement,  where  the  generalisation  refers 
to  the  characters  in  which  all  the  instances  agree.  A 
similar  parallel  may  be  drawn  for  the  instances  and 
their  characters  under  what  we  may  call  the  Joint 
Method  or  Method  of  Complementaries.  Thus  the 
larger  the  number  of  instances  which  agree  in  certain 
characters  relatively  to  the  total  number  of  instances 
observed,  the  higher  is  the  probability  that  some  of 
these  characters  are  dependent  upon  one  another.  And 
the  larger  the  number  of  characters  in  which  certain 
instances  agree  relatively  to  the  total  number  of  charac- 
ters analysed,  the  higher  is  the  probability  that  some 
of  these  instances  will  agree  in  other  characters  besides 
those  analysed. 

§  9.  We  have  said  that  the  probability  of  a  generalisa- 
tion varies  with  the  degree  of  precision  or  determinate- 
ness  with  which  we  are  able  to  define  the  characters  of 
the  instances  examined;  this  determinateness  reaches 
its  highest  point  when  instruments  of  measurement  can 
be  employed;  and  this  accounts  for  the  high  proba- 
bility generally  attributed  to  generalisations  formulated 
in  terms  of  mathematical  concepdons.  Thus  a  further 
criterion  of  probability  rests  upon  the  possibility  of 
applying  quantitative  considerations.  It  may  be  pointed 
out  that  this  criterion  carries  the  one  which  precedes  it 
one  step  further  in  the  direction  of  determinateness: 
the  earlier  dealt  with  the  number  of  characters  belong- 
ing to  different  determinates ;  the  later  deals  with  the 
number  of  determinate  characters  distinguishable  under 
the  same  determinable. 


i 


28 


CHAPTER  II 


§  lo.  The  last  criterion  to  be  mentioned  virtually 
sums  up  and  further  organises  all  the  preceding  criteria. 
It  may  be  said  to  rest  upon  the  comprehensive  com- 
plexity with  which  a  law  correlating  the  examined  in- 
stances can  be  formulated.  The  generalisation  of  course 
consists  in  extending  such  a  formula  to  unexamined 
instances;  and  if  we  have  been  able  to  define  with 
comprehensive  exactitude  the  kind  of  complexity  ex- 
hibited in  the  instances  compared,  then  the  proba- 
bility with  which  the  formula  may  be  extended  to 
unexamined  instances  is  commonly  held  to  approach 
very  nearly  to  certitude. 

§  1 1.  In  discussing  the  general  nature  of  the  method 
of  problematic  induction,  the  relations  of  agreement  and 
difference  are  those  which  have  figured  throughout  as 
the  two  forms  of  analytic  comparison.  From  this  point 
of  view  there  can  be  no  method  of  direct  induction  which 
might  not  be  denominated  by  Mill's  phrase,  the  Joint 
Method  of  Agreement  and  Difference.  It  may  be  use- 
ful, therefore,  to  draw  up  a  two-dimensional  scheme 
corresponding  to  the  two  relations  of  agreement  and 
difference,  which  will  enable  us  to  figure  this  method 
in  imagination.  Instead  of  speaking  of  instances  under 
the  method  of  agreement  or  difference,  we  shall  speak 
of  instances  under  the  relation  of  relevant  agreement 
or  of  relevant  difference.  Instances  which  relevantly 
agree  will  be  figured  in  a  set  of  parallel  columns;  and 
instances  which  relevantly  differ,  in  a  set  of  parallel  rows. 
It  will  be  assumed  that  of  the  instances  schematised, 
those  which  relevantly  agree  with  one  another  will  have 
irrelevantly  differed  from  one  another  as  much  as 
possible;  and  those  which  relevantly  differ  from  one 


1 


i 


4^ 


> 


T 


.^ 


-*   4 


THE  CRITERIA  OF  PROBLEMATIC  INDUCTION      29 

another  will  have  irrelevantly  agreed  with  one  another 
as  much  as  possible. 


(abcd)u  V  w ,,. 
{abcd)u  z/'-ze/... 
{abcd)u'  i/'w ,.. 
{abcd)u"v  7£/... 


(abed)' u  V  w ., 
{abed)' u  v'w'.. 
{abed)'  u'  v"w  . . 
{abed)' u"v  w'.. 


{abed)" u  V  w ... 
{abed)" u  v'  ixf... 
{abed)"  u'  v"'w ,., 
{abed)"u"v  a/... 


In  the  above  table  any  one  of  the  columns  contains 
instances  which  agree  in  the  relevant  characters ^^CZ?, 
while  they  differ  in  some  or  other  of  the  irrelevant 
characters  UVW\  the  column  is  supposed  to  be  in- 
definitely extended  so  as  to  represent  a  collection  of 
instances  presenting  the  largest  possible  variety  as 
regards  the  irrelevant  characters  UVW.  Any  one  of 
the  rows  contains  instances  each  of  which  differs  from 
all  the  others  in  the  relevant  characters,  while  they 
agree  as  far  as  possible  in  all  the  irrelevant  characters. 
It  will  be  noted  that  the  relevant  characters  have  been 
bracketed,  and  the  dashes  affixed  to  the  entire  bracket — 
a  mode  of  symbolisation  which  is  intended  to  denote 
that  the  observer,  being  unable  to  analyse  the  complex 
ABCD  into  its  simplex  factors,  may  have  been  forced 
to  regard  the  variations  as  pertaining  to  the  complex 
as  a  whole.  In  comparing  instances  in  the  same  row, 
therefore,  the  observer  knows  that  there  is  some  differ- 
ence in  the  compared  complexes,  though  he  may  not 
know  to  what  special  factor  within  the  complex  the 
difference  attaches.  If  for  the  bracketed  ABCD,  a 
single  letter — say  M — were  substituted,  it  would  repre- 
sent the  observer's  ignorance  as  to  the  nature  of  the 
factor  he  was  varying — whether  it  was  single,  or  if 
complex,  which  of  the  simplex  factors,  and  how  many 


\ 


30 


CHAPTER  II 


r-- 


of  them  were  being  varied.  Similarly  of  instances  in 
the  same  column,  the  dashes  affixed  to  the  bracket  abed 
denote  that  the  same  variation  is  observed  throughout 
all  the  instances,  although  the  observer  may  not  know 
to  what  simplex  factor  or  factors  it  attaches.  The  dif- 
ferent columns  each  representing  a  set  of  instances  in 
the  relation  of  agreement  constitute  what  I  have  called 
a  set  of  enumeratives;  and  to  represent  symbolically 
the  generalisation  inferred  from  this  aggregate  of  in- 
stances, we  must  imagine  the  columns  extended  on  the 
same  pattern  to  infinity.  The  several  columns  consti- 
tute a  set  of  complementary  enumeratives,  and  are  each 
problematically  extended  into  a  set  of  complementary 
universals;  the  final  generalisation,  representing  each 
value  of  P  as  depending  upon  the  correlated  value  of 
ABCD,  including  all  these  minor  universals. 

§  1 2.  A  scheme  of  this  kind  does  not  of  course  repre- 
sent the  detailed  criteria  used  to  estimate  the  degree  of 
probability  of  different  generalisations;  it  suffices,  how- 
ever, as  a  basis  for  criticising  certain  popular  views  on 
induction.  The  word  'hypothesis'  is  often  loosely  used 
in  this  connection:  all  inductions,  it  is  said,  are  hypo- 
thetical ;  or  again  every  induction  is  based  upon  hypo- 
thesis. These  two  separate  assertions  are  not — as  is 
sometimes  supposed — equivalent.  That  every  induc- 
tion is  hypothetical  presumably  means  that  inductive 
generalisations  must  be  accepted  with  some  reserve  as 
regards  their  probability;  in  short,  that  induction  does 
not  ensure  certitude.  I  nstead  of  speaking  of  induction  as 
hypothetical,  therefore,  I  prefer  to  speak  of  it  as  being 
problema^^;  meaning  by  this  that  inductive  generalisa- 
tions cannot  be  affirmed  with  certitude,  but  only  with  a 


>■ 

1. 


v- 


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4 


i 


r^   ^ 


b.  .1 


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^     * 


-*^ 


*   * 


THE  CRITERIA  OF  PROBLEMATIC  INDUCTION     31 

lower  or  higher  degree  of  probability,  depending  upon 
the  aggregate  nature  of  the  instances  used  to  establish   | 
them.    When  on  the  other  hand  it  is  said  that  every 
induction  is  based  upon  hypothesis,  'hypothesis'  means 
assumption;  and  the  assumption  referred  to  is  some 
such  proposition  as  that  nature  is  uniform.   Thus  if  it  be 
held  that  the  proposition  that  nature  is  uniform  is  not 
certainly  true,  but  only  probably  true,  then  the  degree 
of  incertitude  which  attaches  to  the  uniformity  of  nature 
must    be    attached   to   any    induction    whose   validity 
depends    upon    the    assumption    of   such    uniformity. 
But  again  in  this  case  I  prefer  to  use  the  word  prob- 
lematic; for  what  is  meant  is  that  there  attaches  to 
the  induction  at  least  as  low  a  degree  of  probability 
as  has  been  attached  to  the  proposition  that  nature  is 

uniform. 

A  third  meaning  of  the  word  'hypothesis,'  as  it  is 
used  by  Jevons,  Sigwart  and  Bosanquet  for  example, 
when  they  assert  that  induction  is  based  on  hypothesis, 
requires  separate  discussion.    I  n  particular  Jevons  main- 
tains that  hypothesis  is  the  first  of  the  three  stages  in 
the  completed  inductive  process,  the  second  stage  being 
called  deduction,  and  the  third  verification.    This  use 
of  the  word  hypothesis  to  denote  the  mere  formulation 
of  a  generalisation  which  it  is  proposed  to  establish, 
is,  in  my  opinion,  totally  unjustifiable.    This  so-called 
hypothesis  or  proposal  constitutes  the  first  stage  in  a 
process  of  which  the  third  and  final  stage  is  called 
verification ;  in  these  two  stages,  therefore,  reference  is 
made  to  one  and  the  same  proposition,  which  is  at  first 
propounded   as  ^0  be  proved,  and  finally  asserted  to 
have  been  proved.    In  short  the  relation  between  the 


I 


32 


CHAPTER  II 


two  stages  is  precisely  that  which  obtains  in  Euclid 
between  the  enunciation  of  a  theorem  which  prefaces 
a  demonstration,  and  the  termination  of  the  proof  which 
concludes  with  q.e.d.  But,  if  the  word  Verification'  is 
used  here  in  its  natural  sense  to  imply  that  the  conclu- 
sion of  the  inductive  process  is  'certified,'  then  it  is 
meaningless  to  speak  of  induction  as  hypothetical  in  the 
sense  of  problematic.  Jevons,  however,  maintains  both 
these  views :  namely  that  completed  induction  ends  with 
verification,  and  that  induction  involves  an  application 
of  the  theory  of  probability,  thus  rendering  all  generali- 
sations problematic.  If  it  is  urged  that  I  have  taken 
the  term  verification  too  literally,  and  that  all  that  is 
meant  is  that  the  generalisation  is  confirmed,  and  not 
actually  verified,  at  this  final  stage,  then  the  proper 
account  of  the  process  is  simply  that  at  the  first  stage 
a  generalisation  is  accepted  with  a  relatively  low  degree 
of  probability,  and  at  the  last  with  a  relatively  high 
degree  of  probability;  and  there  is  no  need  to  introduce 
the  term  'hypothesis.' 

My  criticism  of  Jevons'  account  of  induction  extends, 
moreover,  beyond  his  use  of  the  notions  of  hypothesis 
and  verification,  to  the  stage  of  deduction  which  inter- 
venes between  them.  According  to  his  analysis,  the 
proposition  formulated  in  the  first  stage  is  taken  as 
major  premiss  in  a  deductive  process ;  the  minor  premiss 
being  supplied  by  observation  or  experimentation.  With 
these  two  premisses  entertained  in  the  mind  as  possible, 
a  conclusion  is  drawn  on  purely  deductive  principles, 
referring  in  general  to  a  single  kind  of  case.  Then, 
either  with  or  without  experimentation,  we  examine  an 
instance  of  the  type  to  which  the  deductive  conclusion 


>^*^, 

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THE  CRITERIA  OF  PROBLEMATIC  INDUCTION     33 

refers:  and  if,  on  comparison,  accordance  is  found  be- 
tween the  conclusion  deductively  reached  and  the  obser- 
vation of  this  specially  contrived  instance,  the  final  stage 
called  verification  is  attained.  In  a  certain  sense,  then, 
this  deductive  procedure  includes  all  the  purely  mental 
part  of  what  Jevons  represents  to  be  a  complete  in- 
ductive process.  Thus  the  relation  between  deduction 
and  induction  in  his  scheme  may  be  represented : 

{^Deduction,)  If  P^  and  P^  are  true,  then  C  would 

be  true; 

{Induction.)  If  C  is  true,  then  P^  is  true; 
where  P-,  and  P^  stand  for  the  premisses  and  C  for  the 
conclusion.  Now  I  wish  to  point  out  that  the  process 
of  inference  from  P^  and  P^  to  C  is  in  accordance  with 
a  demonstrative  principle;  but  that  inference  from  C 
to  Pj  cannot  be  governed  by  a  demonstrative  principle; 
it  follows,  therefore,  that  Jevons'  attempt  to  reduce  the 
principle  of  induction  to  the  principle  of  deduction  is 
vain.  The  explanation  of  this  blunder  is  to  be  found,  I 
think,  in  a  confusion  between  hypothetical  inference  and 
categorical  inference.  The  deductive  process  to  which 
Jevons  refers  is  a  mere  hypothetical  inference,  which 
might  be  written:  the  truth  of  the  premisses  would 
imply  the  truth  of  the  conclusion;  whereas  the  in- 
ductive process  is  a  categorical  inference,  and  might  be 
written :  the  truth  of  the  premiss  does  imply  the  truth 
of  the  conclusion. 

§  13.  Let  us  now  give  credit  to  Jevons  for  the  truth 
which  lies  concealed  in  his  theory.  It  has  been  ex- 
pressed by  Whewell  and  others  in  the  principle  that  the 
sole  test  of  an  inductive  generalisation  is  accordance  with 
facts.  This  principle  I  hold  to  convey  a  truth,  but  only  a 
J  L  in  3 


34 


CHAPTER  II 


r  >:,. 


partial  truth ;  for  in  the  first  place  it  neglects  the  vari- 
able degree  of  probability  to  be  attached  to  a  generali- 
sation based  solely  upon  accordance  with  facts ;  and  in 
the  second  place  it  neglects  the  variable  degree  of 
accordance  which  could  be  attributed  to  the  relation  of 
the  generalisation  to  the  facts.  Combining  these  two 
neglected  considerations,  the  principle  of  problematic 
induction  may  be  restated  in  the  following  form : 

The  degree  of  probability  to  be  attached  to  a  generali- 
sation based  upon  facts  varies  directly  with  the  degree  of 
accordance  between  the  generalisation  and  the  facts. 
This  maxim  does  not,  of  course,  claim  to  be  expressed 
with  mathematical  precision ;  the  whole  problem  of  the 
theory  of  induction  is  to  define  as  precisely  as  pos- 
sible what  is  meant  by  Varying  degrees  of  accordance.' 
Roughly,  however,  the  main  factors  upon  which  such 
accordance  depends  are  the  number  and  variety  of  in- 
stances covered  by  the  formula,  and  the  determinate- 
ness  with  which  the  formula  fits  the  facts. 

1.  With  regard  to  the  number  of  instances,  the 
generalisation  ranges  over  an  infinite  number  of  pos- 

'  sible  cases ;  hence  the  larger  the  number  of  observed 
facts  found  to  conform  with  it,  the  higher  its  degree  of 
accordance — on  the  score  of  mere  number. 

2.  With  regard  to  variety  of  instances,  the  generali- 
sation claims  to  apply  irrespective  of  circumstance; 
hence  the  wider  the  range  of  variety  of  circumstance 
in  the  instances  observed,  the  higher  will  be  the  degree 
of  accordance  of  the  generalisation  with  the  facts — on 

the  score  of  variety. 

3.  With  regard  to  determinateness,  the  degree  of 
accordance  is  high  in  proportion  as  the  generalisation 


\ 


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THE  CRITERIA  OF  PROBLEMATIC  INDUCTION     35 

fits  the  facts  closely  and  precisely.  Thus,  if  a  formula 
is  comparatively  indeterminate,  then  it  cannot  be  said 
to  accord  closely  with  facts,  even  though  it  may  cover 
a  large  range.  For  example,  the  generalisation  that 
bodies  falling  to  the  ground  move  more  and  more 
rapidly  as  they  descend  may  be  confirmed  by  observing 
an  actual  increase  of  velocity,  in  which  case  a  certain 
degree  of  accordance  could  be  said  to  .obtain  between 
the  formula  and  the  facts.  But  if  the  formula  asserts 
that  for  every  second  the  rate  of  movement  increases 
by  approximately  32  feet  per  second,  and  by  measuring 
the  actual  fall  of  bodies  it  is  ascertained  that  the  velocity 
of  their  descent  does  actually  increase  at  this  rate,  the 
degree  of  accordance  in  this  case  may  be  said  to  be 
high on  the  score  of  comparative  determinateness. 


3—2 


CHAPTER  III 

DEPENDENCY  AND  INDEPENDENCY 

§  I.  Agreement  and  difference — the  two  principles 
upon  which  every  method  of  direct  induction  ultimately 
depends — are  notions  which  may  be  further  expounded 
and  more  precisely  defined  by  a  logical  analysis  of  the 
kind  of  proposition  which  directly  expresses  the  data  of 
observation.  Such  a  proposition  assumes  the  form  : 
certain  observed  manifestations  are  characterised  by 
the  descriptive  adjectives  mnpqr,  say.  Now  this  form 
of  proposition  is — in  two  main  respects — different  from 
that  with  which  we  have  been  chiefly  familiarised  in 
logical  teaching.  In  the  first  place,  the  familiar  terms 
of  quantity,  such  as  'all'  or  'some'  are  omitted;  and 
therefore  one  important  aspect  of  induction  is  that  it 
represents  inference  from  a  proposition  concerning 
'certain  cases'  to  a  conclusion  about  'all  cases'.  In  the 
second  place,  the  proposition  expressing  the  data  of 
observation  does  not  distinguish  between  those  charac- 
ters that  define  the  subject  term  and  those  that  define 
the  predicate  term  :  that  is  to  say,  it  does  not  assume 
the  familiar  form  'Everything  that  is/  is  q!  Hence  in 
passing  from  the  proposition  that  directly  expresses  the 
data  of  observation  to  the  proposition  that  expresses 
the  conclusion  inductively  inferred,  two  kinds  of  trans- 
formation occur.  The  first  transformation  is  from 
'certain'  to  'every,'  and  depends  upon  the  condition  of 
variance  amongst  the  manifestations  recorded.     The 


r  * 


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DEPENDENCY  AND  INDEPENDENCY 


37 


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second  transformation  is  from  a  proposition  containing 
no  characterised  subject  term  to  a  typical  proposition 
which  has  a  characterised  subject  term  as  well  as  a 
distinct  characterising  predicate  term.     This  is  equi- 
valent to  splitting  up  the  conjunction  of  adjectives 
mnpqr  into,  say,  pqr  to  constitute  the  characterising 
description  of  the  subject  term,  and  mn  the  characterising 
description  of  the  predicate  term — a  separation  which 
is  rendered  possible  by  distinguishing  amongst  the  de- 
scriptive adjectives  mnpqr,  those  which  are  independent 
of  one  another — viz.  in  our  illustration  pqr — from  those 
which  are  inferred  to  be  dependent  upon  the  former — 
viz.  mn.    Only  by  combining  these  two  transforming 
processes,  therefore,  can  we  infer  from  the  inductive 
premiss  'Certain   manifestations  are  mnpqr ^    the  in- 
ductive conclusion  '  All  manifestations  that  are  pqr  are 
mn\   and  the  two  essential  conditions  required  ard 
(i)  for  the  transformation  from  'certain'  to  'every,'  vari- 
ance of  the  observed  manifestations;  and  (2)  for  the, 
separation  of  the  subject  characters  from  the  predicate' 
characters,  establishment  of  independence  amongst  the 
several  subject  characters. 

Closer  enquiry  into  the  first  stage  of  this  transfor- 
mation shows  that  it  takes  place  before  any  separation 
of  the  subject  from  the  predicate  characters ;  or  rather, 
to  express  the  distinction  more  suggestively,  of  the 
determining  from  the  determined  characters.  Instances, 
collected  on  the  ground  of  manifesting  certain  characters 
in  common,  will  always  have  manifested  other  characters 
differing  from  instance  to  instance.  These  inconstant 
characters  have  been  omitted  or  eliminated  in  our 
summary  description  mnpqr  of  the  data  of  observation; 


i 


38 


CHAPTER  III 


but  it  is  in  virtue  of  these  omitted  characters  that  the 
observed  manifestations  can  be  said  to  have  the  ag- 
gregate nature  of  variancy.  Thus,  in  this  preliminary 
process,  we  conceive  the  characters  which  maintain 
their  cohesion  as  forming  a  constant  combination  which 
is  incapable  of  being  destroyed  by  variations  in  other 
concomitant  characters,  the  cohesion  being  the  stronger, 
the  greater  the  degree  of  this  variation.  This  process 
corresponds  to  the  principle  of  agreement,  which  has 
two  aspects  :  namely,  the  elimination  of  varying  charac- 
ters, and  the  retention  of  a  combination  of  constant 
characters. 

The  result  of  this  first  transformation  may  be  stated 
in  the  form  :  there  is  some  relation  of  dependence 
amongst  the  characters  mnpqr\  and  the  characters  being 
regarded  as  a  dependent  conjunction,  we  are  led  to  the 
second  stage  of  our  enquiry,  viz.  which  of  them  are 
dependent  upon  the  others?  That  is  to  say,  we  have 
next  to  discover  amongst  the  characters  in  the  constant 
conjunction,  those  which  are  independent  of  one  another, 
and  which  therefore  constitute  the  determining  characters, 
by  the  conjunction  of  which  the  others  are  probably 
determined.  Again  we  rely  ultimately  upon  observation 
of  instances,  which,  in  order  to  lead  to  the  separation 
of  the  characters  pqr  as  independent  of  one  another 
from  the  characters  m  and  n  as  probably  dependent 
upon  them  jointly,  must  have  been  of  such  a  nature 
that  wherever  one  alone  of  the  characters  pqr  has 
varied,  then  m  and  n  will  have  been  found  to  vary ;  and 
wherever  all  the  characters  pqr  were  jointly  constant, 
m  and  n  were  found  to  be  constant. 

§  2.    From  this  point  onwards,  our  observations  may 


i 
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DEPENDENCY  AND  INDEPENDENCY      39 

be  conducted  under  so-called  'experimental  conditions' ; 
the  object  of  enquiry  being  to  discover  the  specific 
values  of  P,  Q,  R  which  determine  specific  values  of 
M,  N.     The  special  value  of  experiment  lies  in   the 
completeness  and  accuracy  with  which  it  enables  the 
experimenter  to  define  the  relations  of  agreement  and 
difference   subsisting   between   the    determining    and 
determined  characters  in  each  of  the  several  instances 
observed.     To  secure  the  highest  degree  of  accuracy 
and  completeness,   it  is  generally  necessary  to  have 
recourse  to  experiment  in  the  strict  sense  in  which  it 
implies  that  we  have  been  able  ourselves  to  contrive 
the  instances  observed.  In  such  a  case  the  experimenter 
knows  beforehand  which  characters  can  be  taken,  for 
his  purpose,  as  independent  of  one  another— in  the 
sense  that  he  can  vary  these  at  will,  with  the  assurance 
that  the  others  will  remain  constant— and  which  char- 
acters are  to  be  taken  as  dependent  upon  the  former, 
in  the  sense  that  he  is  awaiting  their  manifestation  in 
ignorance  as  to  whether  they  will  prove  constant  or 
varying,  or  varying  to  this  or  that  degree.     Thus  he 
determines  with  accuracy  the  determinate  values  p,  q,  r 
before  the  result  of  the  experiment  is  known,  and  then 
measures  with  equal  accuracy  the  determinate  values  m 
and  n ;  in  this  way  ascertaining  the  precise  effect  which 
follows  upon  a  precise  cause,  where  previous  to  the 
experiment,  both  the  cause  and  the  effect  were  defined 
with  comparative  indeterminateness.     But  such  exact 
experimentation  presupposes  the  separation  of  the  de- 
pendent from  the  independent  factors  ;    the  dependent 
characters  being  those  whose  determinate  values  the 
experimenter  wishes   to  learn   as  the  result  of  the 


40 


CHAPTER  III 


1. 


experiment ;  while  the  independent  characters  are  those 
the   determinate   values   of   which   he   knows    before 
experimenting.      As  has  often  been  pointed  out,  the 
experimenter   may  of  course    be  mistaken   on  these 
points;  and  while  varying  one  factor,  be  unintentionally 
varying  another,  which  is  causally  dependent  upon  it. 
This  mistake  would  involve  the  assumption  that  certain 
factors  were  independent  which  were  in  reality  dependent  ; 
but  the  mistake  which  I  propose  next  to  examine  is  the 
supposition,  or  rather  inference,  that  certain  factors  are 
dependent,  when  they  are  really  independent.  A  fallacy 
of  this  kind  arises  only  where  it  is  impossible  for  the 
scientist  to  contrive  a  variation  in  the  characters  mani- 
fested in  nature  in  constantly  cohering  groups. 

§  3.    To  illustrate  such  an  incorrect  supposition,  let  us 
suppose  a  variation  in  b  in  two  instances  symbolised 
as    abcdq,    ab'cdq'.     These    concomitances    might    be 
analysed   either   in    the   form    bcd<^aq   and   Ucd^^a^, 
or   in    the  form  abcdr^^q  and  ab^cd^q^     In   the  first 
analysis  a  is  taken  to  be  dependent,   and   only  bed 
independent  of  one  another;    in  the  second  analysis 
abed  are  taken  all  as  independent  of  one  another.    The 
fallacy  that  we  are  considering  is  the  assumption  that 
the  former  is  correct  when,  in  truth,  the  latter  is  correct ; 
that  is,  the  factor  a  has  been  falsely  supposed  to  be 
dependent,   when  in   reality  it   is  independent.    This 
incorrect  analysis  would  lead  us  to  infer,  (i)  since  the 
variation  of  b  alone  entails  no  variation  in  a,  that  for  all 
instances  ed<^a\    and  (2)  since  the  variation  of  b  alone 
entails  a  variation  of  q,  that  for  all  instances  bedr^q, 
bUdr^q\   But  these  two  inferences  could  not  have  been 
made  from  the  correct  analysis  ;    that  is,  we  could  not 


•t'-i 


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DEPENDENCY  AND  INDEPENDENCY      41 

have  inferred  that  q  would  follow  from  bed  for   all 
values  of  yi,  or  that  /  would  follow  from  Ued  for  all 
values  of  A  ;    because,  as  far  as  the  given  instances 
alone  are  considered,  the  causal  factors  include  the 
determinate  value  a  along  with  the  determinate  values 
of  bed.    Again  the  fact  that  A  is  independent  of  ed 
invalidates  the  inference  edr^a,  since,  as  a  matter  of  fact, 
a  might  have  had  any  value  whatever  concomitantly 
with  ed.     These  inferences  show  that  the  employment 
of  the  figures  of  agreement  and  difference  requires  us 
to  select  beforehand  those  characters  that  can  be  prop- 
erly regarded  as  independent  of  one  another,  as  distinct 
from  those  which  are  dependent  jointly  upon  them;  for 
in  both  cases  error  occurs  because  the  symbol  a  is 
placed  on  the  side  of  the  dependent  factors,  when  it 
ought  to  have  been  placed  among  the  independent.    A 
more  general  form  of  exhibiting  this  same  fallacy  is  to 
recognise  independencies  as  being  of  certain  numerical 
orders.    Thus  the  false  analysis  bed^aq  represents  the 
independency  to  be  of  order  3,  when  the  correct  analysis 
abedr^q  shows  it  to  be  of  order  4.  Although  it  is  always 
mistaken  to  assume  an  independency  to  be  of  a  lower 
order  than  is  actually  the  case,  there  is  no  objection 
to  provisionally  assuming  it  to  be  of  a  higher  order  than 
it  actually  is.    In  fact,  the  conclusions  derived  from  the 
employment  of  the  figure  of  agreement  are  those  in 
which  independencies  provisionally  assumed  to  be  of 
order  4  say,  are  proved  to  be  really  of  order  3,  by 
showing  that  one  of  the  supposed  determining  factors 
may  vary  without  affecting  the  value  of  the  determined 
factor,  and  may  therefore  be  eliminated  from  the  deter- 
mining  group.     Another  way  of  expressing  the  fallacy 


42 


CHAPTER  III 


under  consideration,  therefore,  is  to  say  that  an  inde- 
pendency is  represented  as  of  a  lower  numerical  order 
than  is  correct 

It  must  be  noted  that  the  order  of  an  independency 
is  not  absolute,  but  relative  to  the  effect  factor  whose 
variation  is  under  consideration.  Thus  if  we  suppose 
a  further  complication  added  to  our  original  instance, 
this  would  assume  the  form  a'bcdr^p^  which,  by  com- 
parison with  abcd<^pq,  leads,  in  accordance  with  the 
figure  of  agreement,  to  the  elimination  of  a  as  inopera- 
tive upon  /.  This  inference  is  correct  on  the  assumption 
that  as  regards  the  effect  p,  the  factors  abed  constitute 
an  independent  group.  In  respect  of  the  effect/  alone 
therefore,  the  order  of  independence  is  3,  i.e.  bcdr^p ; 
while,  with  respect  to  the  effect  q,  the  order  of  indepen- 
dence was  4,  since  we  could  not  eliminate  a  as  inoperative 
in  determining  q. 


*»  -V 


-  4 


4k 


<'M, 


CHAPTER  IV 


EDUCTION 


§  I.    The  term  eduction  is  chosen  to  describe  the  kind 
of  inference  whichMill  speaks  of  as  from  particulars  to 
particulars.  In  place  of  Mill's  phrase,  I  should  substitute 
inference  from  instances  to  instances,  and  in  using  the 
technical  term  eduction,  I  wish  to  point  out  where  I 
agree  with,  and  where  I  differ  from,  Mill's  view  of  the 
relation  of  deduction  to  induction.    If  we  consider  the 
singular  instantial  proposition  's  \s  pi  it  might  stand 
first  as  a  conclusion  deduced  from  *  Every  m  \^  f  and 
's\^m\  or  secondly  as  a  premiss  which  together  with 
's\sm'  leads  to  the  inductive  inference  ^ Every  m  is/.' 
Now  according  to  Mill,  a  more  ultimate  analysis  of  the 
deductive  or  syllogistic  argument  reveals  it  to  be  founded 
upon  instances  such  as  s^,  s,,  s,,  ...  which,  being  m,  are 
also/,  so  that  the  universal  ^ Every  misp'  contributes 
nothing  to  the  factual  data  upon  which  the  syllogistic 
conclusion  *^  is/'  is  based ;  and  he  leads  the  reader  to 
assume  that  the  single  conclusion  ^^  is/'  is  established 
with  the  same  force  as  the  universal  *  Every  w  is  /' 
from  the  instances  of  5„  s,,  s,,  ...  that  are  /.    At  this 
point,  however,  I  differ  from  Mill,  and  distinguish  the 
type  of  inference  which  from  ^certain  /s  that  are  m  are 
/'  concludes  that  'Every  m  is  /,'  from  the  type  of  in- 
ference  which,  from  the  same  premisses,  concludes  that 
a  further  instance  of  s  that  is  m  is  /.    The  former  is 
called  induction,  and  for  the  purpose  of  distinction,  I 


44 


CHAPTER  IV 


give  to  the  latter  the  name  of  eduction — this  term  in- 
dicating that  yrhe  conclusion  merely  goes  outside  the 
instances  which  constitute  the  premiss,  while  the  in- 
ductive conclusion  extends  to  all  instances  of  an  assigned 
character.  Mill's  statement  that  the  evidence  for  the 
conclusion  *s  \s  p'  is  the  same  as  the  evidence  for  the 
universal  'Every  m  is/'  is  somewhat  hasty;  for  on 
the  surface  it  would  appear  that  from  the  same  evidential 
data  the  single  conclusion  can  be  drawn  with  higher 
credibility  than  the  universal.  For  the  present,  however, 
we  will  postpone  the  question  of  probability,  and  define 
more  precisely  the  nature  of  the  instantial  premisses 
upon  which  both  the  eductive  and  inductive  procedure 
are  based. 

§  2.    In  this  connection  we  have  first  to  criticise  Mill's 
use  of  the  term  'particular'  in  his  analysis  of  the  process  of 
'inference  from  particulars  to  particulars' ;  for  he  appears 
to  assert  that  the  inference  to  's  is  /'  is  based  merely 
upon  premisses  such  as  s^  is  /,  s^  is  /,  ^3  is  p,  where  s^, 
^2,  ^3  might  stand  for  anything  whatever;  much  as  if  we 
argued,  because  fire  is  red,  and  poppies  are  red,  and 
Mr  Webb's  tie  is  red,  that  therefore  the  British  Con- 
stitution is  red.   In  short  he  has  neglected  the  essential 
question  of  the  mediating  conception,  through  which 
we   pass   from   given    instances   that  are  p  to  some 
new  instance.    This  mediating  conception  is  precisely 
equivalent  to  the  middle  term  of  the  syllogism,  and 
I  therefore  here  represent  it  by  the  symbol  m]  only 
when  it  is  known  that  s  is  m,  and  that  w  is  a  character 
common  to  ^1,  ^2,  ^3,  all  of  which  are  characterised  by/, 
can  we  infer  with  any  semblance  of  probability  that  the 
new  instance  s  will  also  have  the  character  /,  which  was 


EDUCTION 


45 


*N. 


\ 


>V- 


y 


*-  J 


»» 


<rr< 


0*  -^ 


the  proposed  predicate  in  our  conclusion.  Thus,  for  in- 
ferring the  proposed  conclusion  's  is  /,'  the  minimum 
of  instantial  data  required  includes  three  propositions ; 
viz.,  's,  is  w,'  's,  is  /,'  and  '5  is  ;;^,'  where  m  and  p  are 
of  the  nature  of  adjectives  or  universals,  while  the  in- 
stances themselves  are  of  the  nature  of  substantives  or 
particulars.  A  simple  example  of  such  a  triad  of  pre- 
misses is  as  follows: 

Mars  is  a  solar  planet. 
The  earth  is  a  solar  planet. 
The  earth  is  inhabited. 
.*.  Mars  is  inhabited. 

Here  the  only  known  point  of  agreement  between 
Mars  and  the  earth  is  that  they  are  both  solar  planets, 
and  from  this  very  slender  relation  of  agreement  we  infer 
with  the  lowest  degree  of  probability  that  Mars  is  in- 
habited, because  we  know  the  earth  to  be  so.  The 
probability  of  this  conclusion  is  strengthened,  the  greater 
the  number  of  characters  in  which  Mars  is  found  to 
agree  with  the  earth ;  e.g.  its  being  near  the  sun,  and 
having  atmosphere  and  vapour.  It  would  be  still  further 
strengthened,  if  other  solar  planets  besides  the  earth 
were  known  to  be  near  the  sun,  to  have  atmosphere 
and  vapour,  and  to  be  inhabited.  The  more  complete 
process  of  eduction  thus  exemplified  may  be  represented 
in  the  following  scheme : 

(i)  J  is-characterised-by/i  and/2  and  .../to, 

(2)  pi  and/2  and  .,.pm  characterise  Si  and  $2  and  ...J«, 

(3)  si  and  Si  and  ...  J*  are-characterised-by  p, 

.  • .  J  is-characterised-by  p. 

Thus,  in  eduction  there  are  three  summary  premisses, 
containing  {a)  the  summary  term  */,  and  p^,..  and  pj 
which  is  adjectival;  and  {b)  the  summary  term  's,  and 


46 


CHAPTER  IV 


EDUCTION 


47 


s^.,.  and  s^*  which  is  substantival;  besides  the  sub- 
stantival term  s  and  the  adjectival  term  /,  which  occur 
in  the  conclusion.  The  mediating  term  'p^  and  /a . . . 
and  /^'  will  be  denominated  the  intensional  middle 
term,  and  the  mediating  term  's^  and  ^2...  and  s^'  the 
extensional  middle  term,  s  and  /  being  respectively  the 
minor  and  major  terms.  The  premiss  containing  s  will 
be  called  the  minor  premiss;  that  containing  /,  the 
major  premiss  ;  and  that  containing  neither  s  nor  /, 
the  mediating  or  middle  premiss.  The  eductive  scheme 
may  be  conveniently  represented  in  the  form  of  a  chain, 
showing  how  the  subject  and  predicate  of  the  conclusion 
are  linked  up  through  the  two  mediating  terms,  thus : 

Minor  premiss        Major  premiss 
Middle  premiss 

where  x  =  characterises,  and  x  =  is-characterised-by. 

§  3.  Previous  logicians  have  rather  awkwardly  con- 
trasted inference  by  analogy  with  inference  by  induction 
— some  regarding  analogy  as  the  basis  of  induction,  and 
others  taking  induction  to  be  the  basis  of  analogy.  In 
what  sense  these  two  terms  are  used  is  not  clear,  except 
that  induction  is  understood  to  depend  primarily  upon 
the  number  of  instances  known  to  be  characterised  by 
a  certain  adjective  ;  while  the  force  of  analogy  depends 
upon  the  number  of  adjectives  that  are  known  to 
characterise  a  certain  instance.  But  it  is  essential  to 
insist  that  neither  by  accumulating  instances  alone,  nor 
by  accumulating  adjectives  alone,  can  any  inference  be 
drawn,  and  that  inference  of  this  type,  by  whatever 
name  it  may  be  called,  is  governed  by  principles  which 
underlie  both  induction  and  analogy — requiring  an  in- 
tensional as  well  as  an  extensional  link.    For  example, 


f 


■r' 


%^- 


W: 

w  ^ 


;a 


V 


I 


I1 


no  mere  accumulation  of  instances  s^y  s^,  j,,  ...  s^  that 
are  /  could  give  any  probability  that  a  new  instance 
s  will  be  /,  unless  s  were  known  to  have  at  least  one 
character  predicable  of  all  these  instances.  And  con- 
versely, no  accumulation  of  characters /i, /a »  .../«  that 
are  predicable  of  s,  could  give  any  probability  that  a 
new  character/  is  predicable  of  ^,  unless/  were  known 
to  be  predicable  of  at  least  one  instance  having  all  these 

characters. 

For  the  purposes  of  developing  the  principles  of 
eduction  in  their  relations  to  probability,  a  fundamental 
distinction  must  be  made  according  to  whether 

(a)  all  the  evidential  data  are  in  favour  of  s  being  /, 
or  {6)  some  of  the  evidential  data  are  in  favour,  and 

others  unfavourable. 
In  case  (a)  the  eductive  process  leads  to  an  inductive  in- 
ference whose  conclusion  is  universal ;  in  case  (d)  to  an 
inductive  inference  whose  conclusion  is  class-fractional. 
The  remainder  of  this  chapter  will  be  limited  to  case  {a)\ 

Now  the  data  favourable  to  the  proposal  s  is  /,  fall 
into  four  heads : 


( 1 )  I ntermediaries  pi  . . . ,  ^1  • . 

sxPiX^iXp 

(2)  Intermediaries  ^1  ... ,  A  • 

(3)  Intermediaries  jTi  ... ,  Wi 

sx^iX^iXp- 

(4)  Intermediaries ^'i  ...,  ^i  > 

sxnx'^ixP- 


such  that 
such  that 
,  such  that 
,  such  that 


These  four  cases  will  be  recognised  as  all  favouring  the 
proposal  s  is/,  because  they  constitute  the  different  ways 
in  which  an  even  number  of  non-characterising  links 

enter. 

1  Case  (^)  will  be  treated  in  the  Appendix  to  Part  III. 


I 


48 


CHAPTER  IV 


y  ^ 


EDUCTION 


49 


§  4.  If  now  all  the  evidential  items  are  of  one  or  other 
of  these  four  kinds,  then  they  all  co-operate  in  strength- 
ening the  probability  that  s  is  p.  Speaking  generally, 
the  larger  the  number  of  data  of  this  kind  that  have  been 
established,  the  higher  is  the  probability  that  s  is  /. 
But  certain  conditions  must  be  fulfilled  in  order  that 
any  apparently  new  evidence  should  actually  strengthen 
the  required  probability.  First  as  regards  the  extensional 
aspect  of  the  evidence.  The  enumerated  set  of  instances 
Sj .,.  s„  will  count  as  n  separate  data,  provided  that  every 
one  of  them  such  as  s„  is  uncharacterised  by  some  of  the 
adjectives  that  characterise  all  the  remainder,  i.e. 

In  other  words,'^the  instance  s„  counts  as  one  additional 
item  of  evidence  provided  that  it  increases  the  variety 
of  the  evidence  ;  which  it  would  fail  to  do  unless  it  had, 
besides  a  certain  nucleus  of  characters  common  to  all 
the  other  instances,  some  character  opposed  to  the 
common  characters  of  the  others.  Secondly,  as  regards 
the  intensional  aspect  of  the  evidence :  The  enumerated 
set  of  adjectives /i,  ^2  •••/^  will  count  as  m  separate 
data,  provided  that  every  one  of  them  such  as/^,  does 
not  characterise  all  the  instances  that  are  characterised 
by  the  remainder,  i.e. 

Hence  an  instance  s„  adds  to  the  weight  of  evidence 
when  it  constitutes  a  variation  upon  the  other  instances ; 
and  an  adjective/^  adds  to  the  weight  of  the  evidence 
when  it  is  independent  of  the  other  characters. 

§  5.  We  are  thus  enabled  to  establish  certain  principles 
regulating  the  strengthening  force  of  evidential  data. 


\i 


In  inferring  from  examined  subjects  that  a  given  subject 
has  a  property  characterising  these,  we  rely  upon  the 
likeness  of  the  new  subject  to  those  adduced,  and  the 
force  of  any  such  new  instance  varies  with  the  degree 
of  resemblance  to  the  adduced  subjects,  and  with  the 
degree  of  unlikeness  amongst  the  examined  subjects 
themselves.  The  more  remotely  the  latter  differ  from 
one  another,  the  stronger  is  the  evidence  that  they  will 
agree  with  the  new  subject  in  further  points,  beyond 
those  in  which  they  are  known  to  agree.  In  other  words, 
the  more  varied  or  non-congruent  with  one  another  the 
accumulated  subjects,  the  stronger  the  evidence  in  favour 
of  a  certain  congruence.  Similarly  with  regard  to  the  in- 
tensional aspect  of  the  evidence ;  the  predicates  /^ . . .  /^ 
which  serve  as  intermediaries  must  be  as  independent 
of  one  another  as  possible,  when  used  as  evidence  for 
establishing  the  dependence  of  a  further  proposed  pre- 
dicate upon  these  given  predicates  taken  in  conjunction. 
Summing  up,  then,  for  establishing  a  proposed  congru- 
ence, the  condition  required  is  non-congruence  amongst 
the  examined  subjects  ;  and  for  establishing  a  proposed 
dependence,  the  condition  required  is  non-dependence 
amongst  the  examined  predicates. 

These  formulae  represent  the  final  inferred  inductive 
conclusion  in  its  two-fold  universality;  such  universality 
being,  on  both  sides,  unlimited.  In  contrast  to  this  un- 
limited universality,  the  evidence  for  such  an  inductive 
conclusion  has  to  be  exhibited  in  terms  of  what  by 
observation  and  examination  is  known.  Thus  when  all 
the  subjects  known  to  be  characterised  hy  p^.^.p^  are 
also  characterised  by  p,  we  infer  inductively  that  all 
subjects  so  characterised,  will  be  characterised  by  /. 

J  L  III  4 


50 


CHAPTER  IV 


'  *». 


And  when  again  all  the  predicates  that  are  known  to 
characterise  s^  ...  s„  also  characterise  Sy  we  infer  in- 
ductively that  all  the  predicates  which  characterise  this 
set  will  also  characterise  s.  What  we  have  here  expressed 
in  the  form  of  a  set  of  subjects  generating  a  subject 
group,  and  a  set  of  predicates  generating  a  predicate 
group,  is  in  effect  equivalent  to  what  was  above  ex- 
plained as  the  intensional  and  extensional  aspect  of  the 
eductive  process. 

§  6.  It  will  be  observed  that  the  instantial  evidence  for 
the  proposition  's  isp'  does  not  point  to  any  one  given 
s  or/,  but  to  any  subject  characterised  by  the  conjunc- 
tion of  predicates /i.../^,  and  any  predicate  charac- 
terising the  conjunction  of  subjects  s^.,.s^.  We  can 
therefore  eliminate  the  explicit  symbols  s  and  /,  and 
consider  only  what  we  have  called  the  intermediary 
premisses,  summed  up  in  the  proposition  s^.,.s„  are 
characterised  by/^ .../;«.  This  proposition  will  be  called 
the  summarised  evidential  datum,  pointing  to  the  con- 
clusion that  any  unassigned  subject  characterised  by 
A  •  •  •  P»^  ^^''  ^^  characterised  by  any  unassigned  predi- 
cate that  characterises  5^ ...  5„.  This  summary  evidence 
contains  mn  atomic  data,  each  additional  subject  and 
each  additional  predicate  in  the  two  conjunctions 
counting  as  one. 

We  now  proceed  to  consider  the  precise  condition 
required  in  order  that  each  of  these  subjects  and  each 
of  these  predicates  shall  count  as  one,  in  the  estimate 
of  the  evidence  before  us.  The  condition  that  the  pre- 
dicate/^, say,  shall  count  as  an  additional  item,  is  that 
there  shall  be  some  subject,  say  a-„,  such  that 

A  Xo-„xAA  •••/;«-!>  / 


f 


.    i 


~  rf 


.^     ■^ 


EDUCTION  51 

where  there  is  also  a  subject  (r„_i  such  that 

and  so  on.  In  other  words,  each  additional  predicate, 
Pmipm-ii  etc.,  must  be  known  to  be  independent  of  the 
conjunction  of  the  remaining  predicates.  In  this  case  the 
set  of  predicates  A  .../^  will  be  called  an  independency. 
A  corresponding  condition  is  required  for  the  set  of 
subjects  Si.,.s^'y  i.e.  there  must  be  a  predicate  tt^  such 
that 

as  also  a  predicate  tt^.j  such  that 

and  so  on.  In  other  words,  each  additional  subject  s„, 
^^_i,  etc.,  must  differ  in  character  in  at  least  one  predi- 
cate from  the  remaining  subjects.  When  the  set  of 
subjects  satisfies  this  condition,  we  shall  speak  of  it  as 
a  variancy. 

§  7.  Thus  the  parallel  terms  independency  and 
variancy — the  first  applying  to  a  collection  of  predi- 
cates, and  the  second  to  a  collection  of  subjects — can  be 
defined  absolutely ;  i.e.  without  reference  to  any  other 
named  predicate  or  subject.  Now  such  an  independency 
must  be  distinguished  from  another  set  of  predicates 
which  are  invariably  found  to  be  concomitant  with  the 
independent  set  in  our  instantial  evidence.  Up  typifies 
such  a  concomitant  predicate,  />  will  differ  from  any  of 
the  predicates  in  the  independent  set  in  the  point  that 
there  is  no  subject,  such  as  cr,  which  is  known  to  be 
characterised  by  the  set  A---/m»  which  is  known  not 
to  be  characterised  by  /.  Again  the  set  of  subjects 
Si,\,s„y  which  constitutes  a  variancy,  must  be  distin- 

4—2 


52 


CHAPTER  IV 


guished  from  another  set  of  subjects  which  are  found 
to  be  congruent  with  the  set  ^^ ...  ^„  in  all  their  known 
common  characters.  If  ^  typifies  such  a  congruent 
subject,  s  will  differ  from  any  of  the  subjects  in  the 
variant  set  in  the  point  that  there  is  no  predicate  such 
as  TT  which  is  known  to  characterise  the  seti*!  ...^«, which 
is  known  not  to  characterise  s.  There  are  thus  two  com- 
plementary aspects  of  our  evidence,  which  have  been 
summed  up  in  the  negative  form  (i)  that  no  subject 
known  to  be  characterised  by  /i .../;«  is  known  not  to 
be  characterised  by/;  and  (2)  that  no  predicate  known 
to  characterise  s^.,.  s„  is  known  not  to  characterise  s. 
These  two  negatively  formulated  conditions  may  be 
otherwise  expressed  affirmatively;  viz.  first  that  every 
subject  that  is  known  to  be  characterised  by/^.../^ 
is  also  characterised  by  / ;  and  secondly  that  every 
predicate  that  is  known  to  characterise  j'j...^^,  also 
characterises  5.  These  conditions  in  our  ascertained 
knowledge  are  equivalent  to  the  statement  that  all  our 
evidence  is  unexceptionally  favourable  towards  the  con- 
clusion ^  is  /;  i.e.  the  evidence  comes  under  our  first 
heading  (a).  Where  s  is  unexceptionally  congruent  with 
the  collection  ^j . . .  5„  as  far  as  our  knowledge  reaches, 
we  are  led  to  the  inference  that  it  will  be  congruent  with 
this  set  in  all  other  unknown  as  well  as  known  predi- 
cates. Again  where/  is  unexceptionally  concomitant 
with  the  set/i .../;«  as  far  as  our  knowledge  reaches,  we 
are  led  to  the  inference  that  it  will  be  concomitant  with 
this  set  in  all  other  unknown  as  well  as  known  subjects. 
§  8.  Having  brought  out  the  possible  parallels  be- 
tween substantives  and  adjectives,  or  subjects  and  pre- 
dicates, as  far  as  analogy  permits,  we  have  arrived  at 


> 


EDUCTION 


53 


—      * 


4' 


r 


A 
J 


I'i 


[{< 


i. 


a  point  where  an  irreducible  contrast  between  these  two 
categories  prevents  the  further  formulation  of  the  inten- 
sive and  extensive  aspects  of  eduction  on  precisely  the 
same  lines.  The  substantive  or  subject  of  the  simple  pro- 
position 'i  is/'  is  ultimately  identified  and  distinguished 
from  the  adjei:tive  or  predicate  by  a  definite  type  of  act 
of  thought.  The  subject  s  is  definitive  and  nameable,  as 
well  as  identifiable  and  distinguishable  by  such  extrinsic 
relations  as  temporal  and  spatial  position  ;  such  identi- 
fication may  therefore  be  said  to  be  determined  by  an 
act  of  separation,    On  the  other  hand  the  predicate  / 
standsfor  an  adjective  or  character  predicable  of  different 
substantives  by  means  of  an  act  of  comparison  or  dis- 
crimination ;  predication,  therefore,  may  be  said  to  be 
determmed  by  an  act  of  discrimination. 

§  9.    My  whole  philosophical  attitude  depends  upon 
the  recognition  of  a  fundamental  distinction  between 
these  two  types  of  acts,  specially  denominated  separation 
and  discrimination— a  distinction  which  corresponds  in 
my  general  view  to  that  between  the  particular  and  the 
universal.    Inasmuch  as  substantival  separation  is  a  ne- 
cessary preparatory  step  to  identification  or  comparison 
of  character,  separation  may  be  said  to  be  prior  to  dis- 
crimination ;  and  this  entails  a  further  important  contrast 
between  subject  and  predicate.   Thus  there  is,  amongst 
substantives,  nothing  corresponding  to  the  distinction 
and  relation  which  obtains  amongst  adjectives  between 
determinables  and  determinates ;  substantives  proper, 
i.e.  existents,  are  necessarily  disdnci  just  because  they 
occupy  positions  in  the  same  totality  of  time  and  sfjace ; 
whereas  determinates  which  are  o/>/>osed,  are  those  which 
belong  to  the  same  determinable. 


\ 


>  '*~ 


CHAPTER  V 

PLURALITY  OF  CAUSES  AND  OF  EFFECTS 

§  I.  It  has  become  common  in  modern  logical  works 
to  deny  the  applicability  of  the  plurality  of  causes  where 
scientific  analysis  has  succeeded  in  the  final  formulation 
of  natural  laws.  While  agreeing  in  the  main  with  this 
logical  position,  I  consider  that  the  notion  of  plurality ' 
of  causes  and  effects  is  applicable  where  such  scientific 
analysis  is  incomplete,  and  that  its  discussion  therefore 
has  a  place  in  the  logical  foundations  of  science.  Since 
there  are  various  senses  in  which  the  phrase  is  used,  it 
will  be  convenient  to  arrange  the  discussion  under  a 
number  of  heads  of  discourse. 

The  most  elementary  notion  of  plurality  of  causes 
is  that  which  Aristotle  called  'the  fallacy  of  the  conse- 
quent,' meaning  by  this  what  now-a-days  we  call  the 
fallacy  of  simply  converting  a  universal  affirmative. 
Thus,  to  infer  from  'Every  ^  is  e  that  'Every  ^  is  c^  or 
from  'If-then  e  that  'If-then  c'  is  what  in  modern 
logic  we  designate  as  the  confusion  between  a  proposition 
and  its  complementary  ;  and  this  fallacy  has  practical 
application  where  c^  (say)  stands  for  the  characterisation 
of  a  cause,  and  e^  for  the  characterisation  of  its  effect.  It 
is  obviously  fallacious  to  infer  from  the  manifestation 
of  the  character  e^  that  the  character  c^  will  have  been 
manifested  ;  for  the  effect  e^  may  have  been  due  in  any 
given  instance  to  some  other  cause,  say  c^,  and  this 
possibility  entitles  us  to  speak  familiarly  of  plurality  of 


■  •«►     * 


4 


4 


■K 


%t'. 


Ax 


PLURALITY  OF  CAUSES  AND  OF  EFFECTS         55 

causes  in  cases  where  simple  conversion  is  illegitimate. 
This  first  application  of  the  conception  may  be  expressed 
either  as  an  alternative  of  predications,  or  as  a  con- 
junction of  propositions.   Thus,  starting  with  the  effect 
^.,   the  proposition  expressing  plurality  assumes  the 
alternative  form  ;  'Every  e,  is  either  c^  or  c^  or  c,,  etc.'; 
but  starting  with  the  cause,  it  assumes  the  form  of  a 
conjunction  of  propositions,  namely ;  '  Every  f,  is  e^ 
and  every  c,  is  e,,  and  every  c,  is  e,,  etc'    It  is  usual  in 
the  general  exposition  of  plurality  of  causes  to  employ 
the  alternative  form  of  the  proposition ;  and  this  has  led 
to  the  view  that  it  is  impossible  to  infer  from  the  char- 
acterisation of  an  effect  what  in  any  instance  may  have 
been  its  specific  cause;    for  the  alternative  form  of 
proposition  'Every  e,  is  either  c,  or  c,  or  <r.,  etc'  renders 
it  impossible  when  the  effect  is  characterised  merely  by 
the  character  e^,  to  determine  amongst  the  several 
alternatives  c,,c,,c,,  etc,  which  has  actually  operated  in 
any  given  case.  But  there  is  nothing  in  the  fact  of  such 
plurality  of  causes  as  this  incompatible  with  the  view' 
that  a  sufficiently  precise  characterisation  of  the  effect 
enables  us  to  assign  the  specific  cause  in  any  given 
instance.    For  example,  the  alternative  proposition  can 
be  consistently  held  along  with  the  determinate  pro- 
positions 'Every  ej,  is  f,,'  and  'Every  ej,  is  c;  and 
•  Every  ej,  is  t,,'  etc. ;  and  it  is  important  to  note  that  this 
holds  equally  of  the  sufficiently  precise  characterisation  of 
the  cause.     It  is  a  fundamental  error  common  to  most 
accounts  of  plurality  to  suppose  that  something  is  true 
of  the   relation  of  effect  to  cause,   which  is  untrue, 
mutatis  mutandis,  of  the  relation  of  cause  to  effect. 
There  is  absolute  reciprocity  between  cause  and  effect. 


*» 


56 


^'  t 


CHAPTER  V 


and  insufficient  determinateness  in  the  assignment  of 
either  prohibits  inference,  whether  from  effect  to  cause, 
or  from  cause  to  effect.  The  apparent  want  of  reci- 
procity is  simply  due  to  an  imperfection  of  terminology. 
The  word  cause  is  understood  to  denote  a  completed 
assignment  of  cause-circumstances ;  while  the  term  effect 
is  used  to  denote  an  incomplete  assignment  of  effect- 
circumstances.  An  incomplete  assignment  ^  '  cause  does 
not  enable  us  to  infer  a  determinate  effect,  and  hence, 
in  this  sense,  it  is  not  true  that  the  same  cause  involves 
the  same  effect.  On  the  other  hand,  a  completed  assign- 
ment of  the  effect  does  enable  us  to  infer  a  deter- 
minate cause ;  and  hence  the  statement  that  the  same 
effect  does  not  involve  the  same  cause  is  equally  false. 
§  2.  The  question  which  naturally  next  arises  is  what 
constitutes  a  complete  assignment  of  cause  or  of  effect, 
such  that  we  are  able  to  infer  from  one  to  the  other  ? 
There  is,  in  my  view,  no  general  answer  to  this  question : 
each  case  must  be  treated  ad  hoc.  Complete  assignment 
of  cause  will  be  relative  to  the  more  or  less  arbitrary 
assignment  of  the  effect-character;  and  where  effects 
are  assigned  with  perhaps  equal  degrees  of  indeter- 
minateness,  very  different  degrees  of  determinateness 
might  have  to  be  assigned  to  the  cause  in  order  to 
permit  of  inference.  This  holds  equally  with  regard  to 
the  complete  assignment  of  an  effect.  It  will  help  to 
elucidate  the  problem  to  distinguish  under  the  term 
'cause,'  the  'completed  cause'  from  any  'cause-factor'  ; 
and,  under  the  term  'effect,'  the  'completed  effect'  from 
any  'effect-factor.'  Thus,  relatively  to  any  effect- 
character  e,  we  shall  speak  of  the  conjunction  abc  as 
constituting  the  completed  cause,  when  the  universal 


A-  "— 


a*..: 


« » 


4,t. 


s 


PLURALITY  OF  CAUSES  AND  OF  EFFECTS         57 

proposition  ^ Every  a^^  is  e'  holds.     In  this  case,  the 
several  characters  a,  b,  c,  are  cause-factors  of  the  given 
effect  characterised  as  e  ;  and  the  truth  in  the  doctrine 
of  plurality  of  effects  is  expressed  in  the  statement  that 
a  recurrence  of  a  alone  or  of  b  alone  or  of  c  alone  does 
not  entail  a  recurrence  of  any  the  same  effect-character 
such  as  ^;  in  other  words,  the  mere  recurrence  of 
cause-factc    does  not  ensure  the  recurrence  of  any  th 
same    effect-character.     When    then    we   speak    of    a 
completed  assignment  of  the  cause,  we  mean  simply 
such  an  assignment  as  will  ensure  identity,  in  its  several 
recurrent  manifestations,  of  some  character  in  the  effect. 
Similarly  in  the  case  of  effect:  here  we  start  with  some 
cause-character  a,  and  proceed  to  establish  some  con- 
junction of  effect-characters,  say  pqr,  such  that  wherever 
the  conjunction  pqr  is  manifested,  we  can  infer  the 
causal    operation    of  a\    the  conjunction  pqr  is  then 
denominated  the  'completed  effect'  of  a.    On  the  other 
hand,  the  several  characters  /,  q,  r,  are  effect-factors 
from  neither  of  which  alone  could  the  operation  of  a 
have  been  inferred.     This  last  fact  expresses  the  truth 
inherent  in  the  doctrine  of  plurality  of  causes.     From 
the  effect-factor  p  alone,  or  q  alone,  or  r  alone,  we  could 
not  have  inferred  any  such  determinate  causal  factor  as 
a\    while    from    the    conjunction  of  these  characters, 
which  we  have  called  the  completed  effect,  the  character 
a  of  the  cause  may  be  safely  inferred. 

§  3.  To  bring  out  the  fundamental  principle  that 
when  any  invariability  of  relation  can  be  established,  it 
is  always  a  determinate  conjunction  of  characters  from 
which  some  other  character  can  be  inferred,  we  will 
symbolise  the  completed  cause  of  /  as  abed  where 


w  *- 


58 


CHAPTER  V 


PLURALITY  OF  CAUSES  AND  OF  EFFECTS 


59 


*  Every  abed  is  p J  and  the  completed  effect  of  a  as  pqr 
where  'Every  pqr  is  aJ  We  do  not  expect  in  general 
to  be  able  simply  to  convert  these  uniformities  in  the 
forms  'Every/  is  abed'  or  'Every  a  is  pqr']  nor,  from 
the  universal  'Every  abed  is  p'  do  we  expect  to  be  able 
to  infer  that  every  p  is  a,  or  that  every  p  is  <J;  or  from 
the  universal  'Every  pqr  is  a'  that  every  a  is  /,  or 
every  a  is  q,  etc.  ;  for  any  given  case  of  /  might  have 
been  a^b^ed,  and  any  case  of  a  might  have  been  p^q'r. 
So  from  any  one  effect-character  /  we  should  not  in 
general  be  able  to  infer  any  one  of  the  cause-factors 
which  together  constitute  its  completed  cause;  and  from 
'any  one  cause-character  a,  we  should  not  in  general  be 
able  to  infer  any  one  of  the  effect-characters  which 
together  constitute  its  completed  effect. 

For  brevity's  sake  we  will  here  replace  the  compound 
symbol  abed  by  the  letter  x,  and  the  compound  symbol 
pqr  by  the  letter  2,  We  then  have  the  two  universal 
propositions  'Every ji:  is/'  (where/  is  an  effect-charac- 
ter, and  X  its  completed  cause),  and  'Every  2  is  a' 
(where  ^  is  a  cause-factor,  and  2  its  completed  effect). 
As  we  have  already  pointed  out,  the  doctrine  of  plurality 
may  be  expressed  in  two  ways— either  in  the  alternative 
form  of  predication,  or  in  the  conjunctive  form  ;  and  so 
the  above  discussion  may  be  summarised  either  in  the 
alternative  form  'Every/  is  x,  or:r',  or^r^  etc'  and 

*  Every  a  is  2,  or  2\  or  5",  etc'  or  in  the  conjunctive  form 

*  Every  ;tr  is/,  and  every  ;r'  is/,  and  every  :t:''  is/,  etc' 
and  'Every  2  is  a,  and  every  2"  is  a,  and  every  2''  is  a, 
etc'  Colloquially  expressed  these  propositions  are 
embodied  in  the  statements  that  '  in  different  instances 
different  causes  point  to  the  same  effect ' ;  and  that  '  in 


\ 

•^ 


different  instances  different  effects  point  to  the  same 
cause.'  Our  immediate  problem,  then,  is  to  examine 
further  into  the  siofnificance  of  the  term  'different,'  and 
to  enquire  into  its  adequacy  for  expressing  the  true 
meaning  of  plurality. 

§  4.  When  we  assert  that  under  different  antecedent 
conditions  the  same  consequent  is  manifested,  we  may 
be  understood  to  be  simply  asserting  that  any  such  pro- 
position as  'Every  xis p'  means  that/  can  be  predicated 
in  all  cases — differing  in  an  indefinite  number  of  charac- 
teristics from  instance  to  instance — where  the  character ;»; 
is  manifested.  But  the  mere  difference  of  circumstances 
in  such  cases,  being  followed  by  the  same  effect,  cannot 
be  said  to  constitute  plurality  of  causes ;  for,  if  a  charac- 
ter, expressed  by  the  determinable  V  say,  varies  quite 
indefinitely  without  affecting  the  character  /,  then  the 
various  values  of  V  which  may  be  manifested  from 
instance  to  instance  are  simply  eliminated  in  the  pro- 
position that  'Every;*:  is/.'  In  order  to  speak  properly 
of  plurality,  the  case  must  be  such  that  some  but  not  all 
of  the  possible  values  of  a  determinable  can  be  substi- 
tuted in  the  universal  proposition  ;  for  instance  where, 
X  being  the  determinable,  'Every  x  is  /,'  'Every  x!  is 
/,'  'Every ;i:''  is/,'  are  the  only  three  values  of  X  which 
yield/.  The  conception  of  plurality,  therefore,  requires 
us  to  refer  to  a  restricted  range  of  alternatives ;  some, 
but  not  all,  values  of  X  point  to  /  ;  some,  but  not  all, 
values  of  Z  point  to  a.  Further  x,  x\  x'\  it  will  be 
observed,  are  contraries,  opponents  or  disjuncts,  and  so 
the  term  'different'  in  this  account  has  an  added 
significance.  A  rough  example  will  illustrate  this  point: 
death  is  sometimes  caused  by  poison,  sometimes  by  a 


t 


6o 


CHAPTER  V 


blow  on  the  head ;    the  mere  difference  between  these 
two  causes  does  not  bring  out  the  full  significance  of 
plurality,  for  poison  without  the  blow,  or  the  blow  with- 
out poison,  would  produce  the  same  effect  death,  and 
these  are  contrary  circumstances.     A  similarly  rough 
illustration  of  plurality  of  effects  might  be  taken  from  the 
fact  that  both  a  picture  and  a  pattern  point,  as  effects,  to 
human  action  as  cause  ;  here  again  the  full  significance 
of  plurality  is  brought  out  not  merely  by  the  difference 
between  pattern  and  picture,  but  by  the  contrariety 
of  these  effects,  one  being  a  picture   which  is  not  a 
pattern,  and  the  other  a  pattern  which  is  not  a  picture  ; 
while  both  opposed  effects  point  to  the  same  cause, 
namely  human  purpose  or  agency.   In  both  these  rough 
illustrations  where  contrariety  or  opponency  are  substi- 
tuted for  mere  difference,  we  must  emphasise  the  further 
consideration  that  the  range  of  opponency  which  gives 
significance  to  the  idea  of  plurality  is  restricted,  and  is 
therefore  to  be  contrasted  with  a  completely  unrestricted 
range  where  possible  variable  values  would  be  simply 
eliminated.   It  is  not  every  splash  of  colour  that  points  to 
human  purpose,  but,  in  our  assumed  example,  only  those 
which  are  pictorial  or  symmetrical ;  and  it  is  not  every 
condition  of  the  head  or  inner  organs  that  would  point  to 
death,  but  only  some  selected  and  definable  conditions. 
§  5.  As  stated  in  the  previous  section,  each  of  a  number 
of  opponent  cause-characters  x,  x',  x"  (say)  entails  the 
same  effect-character/,  when  conjoined  with  other  cause- 
characters  respectively,;^/,/,/' ...  (say),  ^and  F being 
determinables  indefinitely  variable.    But  it  must  also 
be  recognised  that  each  of  a  certain  Jinite  set  of  values 
x^  x^  x^  (say)  of  Xy  when  conjoined  with  the  same  value 


PLURALITY  OF  CAUSES  AND  OF  EFFECTS         61 

y  of  Fmay  entail  the  same  value/'  of  the  effect-character 
P,   This  fact  is  best  illustrated  by  a  graph  in  which  the 
abscissa  represents  the  variable  cause-character,  and  the 
ordinate  the  variable  effect-factor,  and  where— other 
relevant  circumstances  being  unchanged — a  horizontal 
line  meets  the  graph  in  two  or  more  points  ^,  ^,  C,  Z?.... 
If  one  observation  is  represented  at  one  point  A,  there 
is  a  chance — which  however  is  small — that  the  second 
observation  should  be  on  another  point  B  where  the 
horizontal  through  A  meets  the  curve;  but  this  initially 
small  probability  decreases  continuously  with  every  fresh 
instance.    This  familiar  fact  requires  us  to  modify  some- 
what our  formulation  of  the  figures  of  Demonstrative 
Induction.  Thus,  as  regards  the  figure  of  Agreement,  it 
was  said  that  if  any  two  values,  say  a^  and  a^,  of  the  cause- 
factor  A  (all  other  factors,  b,  c,  d,  e  remaining  constant) 
entail  the  same  effect  value/,  then  P  is  independent  of 
A  (under  thecircumstances^^^i^).  But  such  independence 
would  be  falsely  inferred  if  a^  and  a.  were  two  values 
which  yielded  the  same  effect/.  The  probability,  under 
the  ordinary  circumstances  of  experiment  and  observa- 
tion, of  precisely  these  two  values  occurring  is  in  general 
so  small  as  to  be  negligible ;  and  in  order  to  diminish  even 
this  small  probability  more  than  two  values  of  A  should 
be  experimentally  instanced.    A  similar  correction  is 
required  for  the  figure  of  Difference.    Thus,  \{d  and  d' 
have  yielded  different  values/  and  /'  of  P  (under  other- 
wise constant  circumstances)  we  cannot  infer  that  literally 
any  other  value  of  D  would  yield  a  different  value  of  P; 
for  there  is  a  small  chance  that  in  another  instance  we 
might  happen  to  hit  upon  one  of  the  values  of  D  that 
yields  the  same  value/  of  P  as  that  yielded  by  d. 


62 


CHAPTER  V 


* 

§  6.    We  have  attempted  to  prove  that  the  relation 
between  cause  and  effect  is  reciprocal  in  the  general 
sense  that  whatever  is  true  of  cause  to  effect,  will  be 
true  of  effect  to  cause,  whether  the  relation  asserted  is 
in  the  form  of  a  universal,  a  particular,  a  conjunctive, 
or  an  alternative  proposition.    We  now  proceed  to  in- 
vestigate how  the  relation  of  cause  to  effect  can  be 
rendered  reciprocal  in  the  more  precise  sense  in  which 
it  involves  the  conjunction  of  the  two  complementary 
universal  propositions  'Every  C  is  E'  and  'Every  E 
is  C!    Now  in  formulating  any  universal  proposition,  we 
begin  by  taking  as  predicate  term  an  arbitrarily  assigned 
character  which  may  stand  either  for  a  cause-character 
or  for  an   effect-character.    We  then  attempt  to  find 
some  conjunction  of  characteristics  from  the  manifesta- 
tion of  which   the  arbitrarily  assigned  character  may 
be    inferred.     This    conjunction   of  characteristics  is 
denominated  'the  completed  cause'  when  the  original 
character   stands  for  an   effect;    and  is  denominated 
the    'completed   effect'    when   the    original    character 
stands  for  a  cause.     Thus,  in  general,  starting  with  the 
predicate  term/,  we  attempt  to  establish  such  a  propo- 
sition as  'Every  abed  is  /.'    If,. at  this  point,  we  turn 
from  p  to  abed,  and  attempt  to  find  a  universal  mark 
from  which  the  conjunct  character  abed  could  be  in- 
ferred, we  discover  a  mark  definable  by  some  such  con- 
junction 2Spqr,  We  should  then  have  the  two  universals 
'Every  abed  is  /'  and  'Every /^r  is  abed,'  where  abed 
defines  the  completed  cause  of  /,  and  pqr  the  com- 
pleted effect  of  abed.    The  next  step  in  our  approxi- 
mation to  a  reciprocal  universal  is  to  attempt  to  find 
the  completed  cause  oi pqr.    Since  abed  constitutes  the 


Ai 


PLURALITY  OF  CAUSES  AND  OF  EFFECTS         63 

completed  cause  of  p,  the  completed  cause  of  pqr  will 
include  abed,  but  it  may  require  further  determination : 
abedef,  let  us  suppose,  is  found  to  be  the  completed 
cause  oipqr.   By  this  process  we  approximate  more  and 
more  closely  to  a  reciprocal  universal,  which  may  ulti- 
mately  be  supposed  to  assume  the  form  'Every  abedep^ 
is  pqrs,  and  every  pqrs  is  abedef!   Let  us  retrace  the 
steps  by  which  we  arrive  at  this  double  formula :  be- 
ginning with/  alone,  we  established  *  Every  abed\sp'\ 
next  with  the  conjunction  abedd^s  our  starting-point,  we 
established  the  universal  '  Every /^r  is  abed' ;  then  with 
the  conjunction/^^ as  our  starting-point  we  established 
the  universal  'Every  abedef\spqr\  and  lastly,  starting 
with  the  conjunction  abedef^^  established  the  universal 
'  Every /^r^  is  abedef!    This  procedure  is  assumed  to 
have  reached  its  termination  from  the  fact  that  abedef 
implies  not  only  pqr,  but  also  s,  so  that  the  relation  of 
inferability  between  abedef  on  the  one  hand  and  pqrs 
on  the  other  hand  is  reciprocal. 

§  7.  The  notion  of  a  completed  effect  or  a  completed 
cause  may  be  approached  from  another  point  of  view. 
Taking  as  before  abed  to  be  typical  of  a  cause-con- 
junction, we  shall  enquire  what  effect-characters  can  be 
inferred  wherever  this  cause-conjunction  is  manifested. 
Let  us  suppose  that/,  q,  and  rare  three  independently 
definable  characters  which  can  be  called  effects  of  the 
junction  abed,  so  that  the  universal  proposition  'Every 
abed  is  pqr  can  be  asserted.  But  if  /,  q,  r  are  to  be 
called  effects  proper  to  the  cause-conjunction  abed,  a 
further  condition  beyond  the  truth  of  the  universal  pro- 
position is  required.  Not  only  must  it  be  true  that 
*  Every  abed  is  pqr,'  but  it  must  also  be  true  that  neither 


64 


CHAPTER  V 


PLURALITY  OF  CAUSES  AND  OF  EFFECTS 


65 


p,  nor  q^  nor  r  could  be  inferred  as  effect  from  any 
cause-conjunction  involving  some  only  of  the  factors 
a,  b,  c,  d.    For  example,  if  every  abc  were  p,  then  p 
would  be  not  an  effect  proper  to  abed,  because  /  would 
be  the  effect  proper  of  the  part-conjunction  abc.    Thus 
in  order  to  find  the  effects  proper  to  the  conjunction  abed, 
we  must  exclude  all  effects  which  could  be  inferred  from 
a  alone,  or  from  ab  alone,  or  from  be  alone,  or  from  abc 
alone,  etc.    Or,  to  take  another  illustration  of  a  closely 
connected  point,  the  universal  proposition  *  Every  abed 
is  x'  would  not  be  a  true  expression  of  causal  law  sup- 
posing that  we  could  have  dropped  the  d,  and  expressed 
the  universal  in  the  wider  form  'Every  abc  is^/    Rela- 
tively to:r,  the  conjunction  a^^^  would  contain  a  super- 
fluous factor,  and  super-complete  assignment  of  cause 
is  as  invalid  as  insufficient  assignment;   thus,  in  for- 
mulating a  universal  proposition  stating  a  causal  rela- 
tion, the  cause  must  not  only  be  complete,  but  it  must 
not  be  super-complete.    These  conditions  are  effected, 
in  the  case  before  us  for  instance,  by  taking  separately 
each  of  the  cause-characters  a,  b,  e  and  d,  and  finding 
the  effects  which  are  due  first  to  the  factors  taken  one 
by  one;  secondly  to  the  factors  taken  two  by  two;  and 
thirdly  to  the  factors  taken  three  by  three;  and  thus 
finally  to  reserve  as  the  effects  proper  to  the  conjunction 
abedxhos^  which  can  be  inferred  from  the  complete  con- 
junction alone.  Assuming  then  that  we  have  standardised 
our  causal  formula  by  excluding  those  effects  for  which 
abed  would  be  a  super-completed  cause,  we  shall  sup- 
pose that  /,    q,  r  severally  are  effects  proper  to  the 
completed  cause  abed.    The  question  next  arises  as  to 
whether />^r  is  the  completed  effect  of  abe\  for  just  as 


I  • 


T 


V 


4 


in  completing  the  assignment  of  cause  we  have  to  avoid 
the  error  of  redundancy,  so  also — though  for  a  different 
reason — we  have  to  avoid  redundancy  in  our  assignment 
of  the  completed  effect.  The  super-completed  effect 
will  be  one  in  which  we  have  failed  to  distinguish  an 
effect  from  the  effect  of  an  effect ;  thus  we  should  have 
wrongly  assigned  pqr  as  the  effect  of  abed,  if  r  itself 
were  the  effect  of  pq.  In  other  words,  the  completed 
effect  must  consist  in  an  independent  conjunction.  Sum- 
marising the  conditions,  then,  for  the  correct  formula- 
tion of  the  causal  law  which  presents  abed  as  the  cause 
of  pqr: 

(a)  The  characters  a,  b,  c,  d  must  be  independently 
definable  and  independently  co-variable. 

(b)  The  characters/,  q,  r  must  also  be  independently 
definable  and  co-variable. 

(c)  None  of  the  effects/,  q,  r  must  be  inferable  from 
a  conjunction  included  in,  but  less  comprehensive  than, 
the  conjunction  abed\  and  conversely. 

If/,  q,  r  are  the  only  effect-characters  which  satisfy 
these  conditions,  then  the  conjunction /^r  may  be  called 
the  completed  effect  of  the  cause-conjunction  abed.  Any 
other  effect-character  such  as  s  would  have  to  be  ex- 
cluded, either  because  it  was  an  effect  of  pqr,  or  be- 
cause it  was  an  effect  of  a  conjunction  more  compre- 
hensive or  less  comprehensive  than  abed.  Finally  then, 
when  the  above  conditions  are  satisfied  the  relation 
between  the  cause-conjunction  abed  and  the  effect-con- 
junction pqr  is  reciprocal;  so  that  *  Every  abed  is  pqr 
and  'Every/^r  is  abed'  \  moreover,  both  uniqueness  of 
effect  entailed  by  the  given  cause,  and  uniqueness  of 
cause  entailed  by  the  given  effect  are  secured. 

J  L  III  ,  5 


CAUSE-FACTORS 


67 


CHAPTER  VI 


CAUSE-FACTORS 


§  I.    The  validity  of  the  antithesis  between  nomic 
necessity  and  universality  of  fact  being  admitted,  it  has 
frequently  been  supposed  that,  within  the  range  of  the 
nomically  necessary,  causal  laws  can  be  distinguished 
from  non-causal  laws.    But  this  view  must  be  rejected. 
Causal  laws  have  been  held  to  apply  only  where  change 
is  involved;  we  have  therefore  to  enquire  into  the  sig- 
nificance of  this  notion,  and  in  place  of  the  somewhat 
obscure  term  change,  I  shall  introduce  the  notion  of 
alterable  as  opposed  to  unalterable  states  of  a  thing. 
This  phraseology  would  not  be  admitted  by  those  philo- 
sophers who  recognise  only  events  or  occurrents,  and 
do   not  allow,   except  for   linguistic  convenience,  the 
notion  of  a  continually  existing  thing  to  which  states 
or  occurrents  are  referable.    I  must  here  restate  in  more 
detail  my  view  that  any  occurrent  is  to  be  referred  to 
a  continuant,  and  that  the  relation  of  an  occurrent  to 
its  continuant,  or  inversely  of  a  continuant  to  .any  of 
its  several  occurrents,  is  a  unique  relation,  to  which 
there  is  no  analogue  in  any  other  aspect  of  reality. 
A  relation  sometimes  hastily  confounded  with  that  of 
occurrent  to  continuant — which  I  will  call  inherence — 
is   the    relation   of  substantive  to  adjective,  which  I 
call  characterisation.     But  since  an  occurrent  may  be 
variously  characterised  it  is  obvious  that  it  stands  to  its 
characterisation  as  substantive  to  adjective;  the  relation, 


'I  '1 


& 


I'/ 


therefore,  of  the  occurrent  to  its  characterisations  can- 
not be  identified  with  the  relation  in  which  it  stands 
to  its  continuant.  The  continuant  itself  might  be  called 
a  substantive  proper,  in  the  narrowest  possible  sense  of 
this  phrase ;  but  I  include  under  the  phrase  substantive 
proper  both  the  occurrent  and  the  continuant,  thereby 
indicating  that  the  relation  of  the  one  to  the  other  is 
not  the  same  as  that  of  substantive  to  adjective. 

To  define  more  explicitly  the  notion  of  a  continuant, 
we  will  assume  that  any  continuant  has  several  modes 
of  existence,  or  rather  modes  of  manifestation  of  exist- 
ence, each  of  which  may  theoretically  be  conceived  as 
a  determinable';  and  according  to  the  nature  of  this 
set  of  determinates,  the  continuant  may  be  said  to 
belong  to  one  or  another  category.  We  assume  further 
that  during  the  period  throughout  which  a  continuant 
exists,  every  one  of  its  modes  is  being  manifested  in 
some  or  other  of  its  determinate  forms.  I  n  the  proper 
mathematical  sense,  time  is  of  one  dimension,  but  in 
order  to  conceive  of  the  existence  of  a  single  con- 
tinuant, it  will  be  helpful  to  represent  time,  in  a  sort 
of  figurative  imagery,  as  having  a  number  of  parallel 
dimensions.  Applying  this  figure  of  speech  to  the 
continuant,  we  may  say  that  its  existence  is  prolonged 
along  a  number  of  parallel  lines  of  time,  each  of  which 
manifests  from  moment  to  moment  the  several  modes 
of  manifestation  in    one  or   other  determinate   form. 


1  My  terminology  should  be  compared  specially  with  that  of 
Descartes  and  Spinoza.  What  I  call  a  determinable  is  almost 
equivalent  to  what  they  call  an  attribute,  and  my  determinate  almost 
equivalent  to  their  mode  of  an  attribute.  My  use  of  the  term  '  mode ' 
will,  therefore,  be  seen  to  differ  from  theirs. 

5—2 


68 


CHAPTER  VI 


CAUSE-FACTORS 


69 


These  lines  of  time,  therefore,  are  conceived  as  being 
completely  filled  or  occupied  by  actual  manifestations ; 
and  the  conception  of  parallel  time-lines  must  be  ex- 
tended so  as  to  apply  to  all  continuants. 

§  2.  With  these  preliminary  remarks  I  will  pass  to  the 
temporal  and  spatial  relations  involved  in  the  conception 
of  causality.  In  the  first  place  the  antithesis  between 
occurrent  and  continuant  corresponds  to  the  antithesis 
between  the  transient  and  the  permanent  or  persistent. 
Popularly  speaking,  what  exists  may  have  only  tran- 
sient existence,  or  else  persist  continuously  throughout 
a  period  of  time,  perhaps  indefinitely  prolonged  at  both 
ends.  In  attributing  continued  existence  to  a  thing,  we 
do  not  mean  that  some  property  of  the  thing  continues 
unchanged ;  for  a  property  stands  to  its  continuant  in 
the  relation  of  adjective  to  substantive.  There  is  a 
further  distinction  amongst  properties  which  charac- 
terise a  continuant,  according  as  these  change  or  persist 
unchanged  throughout  a  period  of  time.  The  continuity 
of  the  existent  is  something  behind  even  the  possibly 
changing  properties,  and  change  applies  not  to  the  con- 
tinuant itself,  but  to  the  adjectives  which  characterise 
it  or  its  occurrents.  Often  the  term  cause  is  applied  in- 
discriminately either  to  the  continuant  itself  or  to  some 
of  its  properties  regarded  as  permanent  in  relation  to 
the  particular  occurrences  or  events  as  effects.  Cause, 
in  this  sense,  is  essentially  something  persisting  through- 
out time,  and  effect  something  essentially  transient  and 
alterable;  so  that  the  cause  is  not  homogeneous  with 
the  effect,  and  this  usage  of  the  term  cause  is  to  be 
carefully  distinguished  from  the  notion  as  applied  to 
related  occurrences.    Mill  fails  to  point   out  this  dis- 


ii  <. 


At 


tinction  when  he  allows  himself  to  deal  in  a  separate 
chapter  with  permanent  causes  or  cause-agents,  and  in 
so  doing  departs  entirely  from  his  preliminary  account 
of  cause  and  effect  as  temporally  related,  the  one  as 
antecedent  and  the  other  subsequent  in  time.  We  must 
be  on  our  guard,  then,  against  the  habit  of  confusing 
causality  regarded  as  a  relation  between  events  with 
causality  regarded  as  the  relation  of  a  permanent  ex- 
istent to  its  alterable  conditions  or  relations.    For  the 
present  I  propose  to  confine  the  discussion  to  the  more 
common  and  familiar  application  of  this  notion  to  oc- 
currences.    In  this  sense  cause  and  effect  are  homo- 
geneous; i.e.  the  same  sort  of  thing  that  can  be  said 
about  the  relation  of  cause  to  effect  can  also  be  said 
about  the  relation  of  effect  to  cause.    Thus  if  a  cause 
process  is  simultaneous  with  an  effect  process,  this  tem- 
poral relation  of  simultaneity  is  convertible ;  or  again 
if  the  cause  process  is  anterior  to  the  effect  process, 
the  latter  is  posterior  to  the  former.    There  is  a  further 
reciprocity  between  cause  and  effect  when  we  conceive 
of  objective  determination  in  its  wide  sense;  for  it  is 
held  in  modern  times  that  the  specific  characterisation 
of   an  effect  determines  the  cause  in  the  same   ob- 
jective sense  as  the  specific  characterisation  of  a  cause 
determines  the  effect.    This  view  is  almost  universally 
accepted,  at  any  rate  from  the  epistemic  point  of  view; 
i.e.  it  is  held  that  the  knowledge,  say,  of  a  sufficiently 
omniscient  being  of  what  is  customarily  called  the  effect, 
would  permit  of  inference  as  to  the  nature  of  the  cause 
with  just  as  much  certainty  as  inference  from  the  know- 
ledge of  the  cause   to  the  knowledge  of  the  effect. 
If  this  be  so  a  real  problem   arises   as   to   whether 


agftk 


70 


CHAPTER  VI 


CAUSE-FACTORS 


71 


ontologically,  as  opposed  to  epistemologically  there  is 
any  objective  antithesis  between  the  relation  of  cause 
to  effect  and  that  of  effect  to  cause,  since  each  of  them 
may  be  said  to  determine  unequivocally  the  nature  of 
the  other. 

These  general  considerations  lead  to  an  apparent 
paradox  with  respect  to  the  reference  of  causality  to 
time  and  space.  Philosophers,  scientists  and  logicians 
alike  have  often  put  forward  as  the  one  supreme  prin- 
ciple of  causality,  that  the  causal  dependence  of  event 
upon  event  is  wholly  unaffected  by  temporal  and  spatial 
differences.  On  the  other  hand  the  analysis  of  every 
phenomenon  in  terms  of  cause  and  effect  assigns  spa- 
tio-temporal relations  between  cause  and  effect.  This 
paradox  is  removed  by  considering  that  the  formula  in 
accordance  with  which  one  event  is  causally  connected 
with  another,  is  independent  of  the  date  and  location  of 

I  the  events,  but  dependent  on  the  temporal  and  spatial 
relations  between  them. 

§  3.    The  alleged  distinction  between  two  types  of  ob- 
jective law  serves  to  introduce  Mill's  distinction  between 
uniformities  of  co-existence  and  causal  laws.  The  phrase 
*  uniformity  of  co-existence' requires  special  considera- 
tion, because  it  has  to  be  distinguished  on  the  one  hand 
^  from  formal  universals,  and  on  the  other  from  causal 
'  laws.    Formal  universals  are  concerned  with  the  spatial 
and  spatio-temporal  relations  involved  in  the  notion  of 
movement.    The  difference  between  such  formulae  and 
^  those  which  connect  the  properties  of  continuants  or 
the  characters  of  occurrences,  is  that  the  latter  refer  to 
existents  whereas  the  former  do  not ;  the  term  existent 
here  being  understood  to  apply  to  what  is  potentially 


.<*  ^ 


or  actually  manifested  in  time,  or  in  space,  or  in  both 
time  and  space.    Briefly  the  geometrical  and  kinematic 
formulae  comprised   under  formal  universals  express) 
the  nature  of  time  and   space   themselves;    whereas! 
uniformities  of  co-existence  and  of  causation  express 
the  nature  of  that  which  occupies  time  and  space.   The 
latter  uniformities  therefore  include  or  presuppose  the 
former,  while  obviously  the  former  do  not  include  or 
presuppose  the  latter.    Passing  now  to  the  contrast  or 
connection  between  uniformities  of  co-existence  and  uni- 
formities of  causation,  the  two  points  which  we  shall 
proceed  to  maintain  are,  first,  that  no  causal  law  can 
be  formulated  except  by  reference  to  co-existing  pro- 
perties of  continuants  as  well  as  by  reference  to  change- 
able occurrences;  and  secondly,  that  the  required  dis- 
tinction is  not  simply  one  of  temporal  relation,  such  as 
simultaneity  and  sequence. 

84.    If  we  consider  what  is  involved  in  defining  or 
describing  an  occurrence,  we  find  that  it  must  always 
entail  reference  to  a  continuant ;  and  that  one  occurrence 
is  defined  as  agreeing  with  or  differing  from  another, 
by  reference  to  the  properties  of  the  continuants  con- 
cerned. For  example,  the  occurrence  described  as  drink- 
ing water  is  different  from  the  occurrence  defined  as 
drinking  ether,  not  by  reference  to  anything  which  could 
be  described  in  terms  of  actual  perceptible  phenomena, 
but  by  reference  to  the  different  properties  or  potenti- 
alities implied  by  the  terms  ether  and  water  respectively, 
which  denote  different  kinds  of  continuants.    It  is  true 
that  the  smell  and  taste  of  ether  would  immediately 
distinguish  it  in  sensation  from  water;  but  for  a  person 
who  might  accidentally  have  lost  his  susceptibility  to 


r 


(^ 


U 


V 


72 


CHAPTER  VI 


CAUSE-FACTORS 


73 


smell  and  taste  these  perceptible  differences  would  be 
unnoticed.    Hence  in  considering  the  causal  conditions 
which  produce  the  different  effects  following  upon  the 
taking  of  ether  or  the  taking  of  water,  the  different 
properties  of  these  substances  must  be  specified.  Further 
proof  of  the  inadequacy  of  the  statement  of  causation 
which  regards  the  cause  as  an  actual  occurrence  related 
as  simultaneous  with  or  antecedent  to  the  effect  occur- 
rence,   lies    in    the    fact   that   the   cause   assigned    to 
account  for  a  given  effect  includes  not  merely  what  has 
occurred  in  actuality,  but  what  would  have  occurred 
under  totally  different  circumstances.   Thus  the  cause 
assigned  to  account  for  the  observed  effects  of  drinking 
ether  would  be  that  ether  is  poisonous,  and  this  state- 
ment, though   explicitly  asserting  the  co-existence  of 
certain  properties,  is  implicitly  a  statement  of  causal 
law,  presumably  discovered  by  means  of  experiments 
in  the  laboratory. 

§  5.    Not  only  is  the  cause,  in  my  view,  something 
more  than  a  mere  actual  occurrence,  but  the  effect  may  be 
something  more  than  a  mere  actual  occurrence ;  for  there 
are  many  cases  in  which  a  complete  account  of  the  effect 
must  include  besides  what  could  be  characterised  in  an 
occurrence  as  immediate  and  actual,  a  characterisable 
change  of  property,  i.e.  of  a  potentiality  that  may  be 
actualised  in  future  manifestations.    This  is  most  ob- 
viously illustrated  from  the  phenomena  of  habit  and 
memory ;  for  it  is  in  mind  that  modifiability  of  property 
is  specially  prominent.    Another  respect  in  which  such 
phenomena  differ  from  those  which  are  described  in  an 
ordinary  account  of  an  occurrence  is  that  whereas  an 
occurrence,  taken  as  effect,  is  generally  referred  to  a 


1 


Yi 


\ 


1 


,  f 
I 

4 


tv". 


single  completely  characterised  occurrence  as  cause,  in 
the  case  of  habit,  the  effect  produced  must  be  accounted 
for,  not  by  a  single  previous  occurrence,  but  by  a  re- 
petition of  occurrences  agreeing  with  one  another  in 
some  respect\  These  illustrations  suffice  to  show  first, 
the  necessity  of  referring  an  occurrence  to  a  continuant; 
and  secondly,  the  necessity  of  including  in  an  account 
of  causal  conditions,  properties  defining  the  potentiali- 
ties of  occurrents,  as  well  as  characters  describing  the 
actual  occurrence.  My  terminology  may  be  compared 
with  the  Aristotelean  classification  of  causes ;  for  Aris- 
totle's material  cause  corresponds  closely  to  what  I  call 
the  continuant  whose  nature  is  manifested  in  causal 
processes;  and  his  efficient  cause  approximately  corre-| 
sponds  to  what  I  call  the  property,  and  which,  when  I 
analysed  as  a  potentiality  corresponds  to  the  Greek 
term  Suj/a/it?.  Finally  what  the  scholastic  logicians  term 
the  'occasional  cause'  is  to  be  understood  as  equivalent 
to  the  occurrent  cause.  It  should  be  noted,  however, 
that  such  distinctions  are  incorrectly  described  as  dis- 
tinctions amongst  causes,  for  they  are  really  distinctions 
amongst  causal  factors.  Thus  the  continuant,  the  pro- 
perty of  the  continuant,  and  the  occurrence,  are  three 
factors  which  jointly  constitute  the  completed  account 
of  the  cause.  When  a  causal  law  is  expressed  in  con- 
densed form  as  a  coexistence  of  properties,  it  is  abso- 
lutely essential  that  the  term  used  to  denote  the  cause 
should  not  connote  a  property  which  represents  the 
effect,  for  otherwise  the  supposed  law  is  nothing  but  a 

^  This  may  of  course  be  resolved  into  the  preceding ;  for  each 
single  occurrence  effects  a  change  in  the  potentialities  of  future 
occurrences. 


I 


74 


CHAPTER  VI 


verbal  proposition.  For  example,  the  proposition '  Poison 
always  kills  people'  appears  to  express  a  uniformity  of 
co-existence  between  the  property  defined  as  poisonous, 
and  the  property  of  occasioning  death  whenever  intro- 
duced into  a  living  person;  but  since  this  last  property 
merely  gives  the  meaning  of  the  first,  the  proposition 
neither  expresses  a  genuine  law  of  causation  nor  a 
genuine  law  of  co-existence. 

§  6.  Turning  now  to  our  second  point  of  distinction 
and  connection  between  uniformities  of  co-existence  and 
uniformities  of  causation,  we  must  direct  special  atten- 
tion to  the  occurrent  factors  in  causation,  with  the  object 
of  examining  first  the  temporal  and  later  the  spatial 
relations  between  the  cause-occurrence  and  the  effect- 
occurrence.  The  typical  case  of  causation  which  has 
figured  most  prominently  in  philosophy  since  the  days 
of  Hume,  is  where  the  cause-occurrence  is  taken  to  have 
preceded  in  time  the  effect-occurrence.  The  language 
used  to  express  the  relation  of  temporal  sequence  or 
succession  has  generally  suggested  the  idea  that  the 
cause-occurrence  can  be  dated  at  one  moment  of  time, 
and  the  effect-occurrence  at  another  moment  of  time, 
with  a  temporal  interval  between  the  two  moments. 
Philosophical  criticism  has  generally  rejected  this  ac- 
count on  the  ground  that  it  implies,  as  a  necessary  con- 
dition for  the  existence  of  the  effect,  the  non-existence 
of  the  cause.  In  other  words,  the  time  at  which  the 
cause,  as  here  regarded,  operates  in  determining  the 
character  and  existence  of  the  effect,  is  the  time  at 
which  the  cause  has  ceased  to  exist,  and  can  therefore 
no  longer  manifest  its  character.  This  attack  upon  the 
common  statement  of  causation  has  led  to  an  attempt 


^ 


I 

ft  * 
} 


i  J 


V'*- 


'#* 


,^    «# 


i 


CAUSE-FACTORS 


75 


to  overthrow  the  notion  of  causality  itself,  on  the  ground 
that  it  involves  an  irremovable  paradox  or  contradiction. 
Now  we  have  only  to  apply  the  above  account  of  cau- 
sation to  the  simplest  known  case  of  causal  process  to 
see  that  in  truth  it  is  fallacious.  The  mere  datum  which 
defines  the  collocation  of  a  system  of  particles,  would 
not  enable  us,  even  with  the  completest  knowledge  of 
the  causal  laws  of  motion,  to  assign  their  subsequent 
positions.  The  datum  in  this  case  defines  an  occur- 
rence by  the  position  at  a  moment  of  time  of  each  of 
the  particles ;  but  the  further  datum  required,  in  order 
to  ascertain  the  positions  at  a  subsequent  moment  of 
time,  is  the  rate  of  movement  of  each  particle  within  a 
period  of  time.  This  simple  case  points  to  the  general 
principle  for  defining  the  temporal  relation  between  a 
cause-occurrence  and  its  effect-occurrence.  Instead  of 
dating  a  cause-occurrence  and  an  effect-occurrence  a/ 
two  separated  moments  of  time,  we  must  define  the 
cause-occurrence  as  a  process  going  on  within  a  certain 
period  of  time,  and  the  effect-occurrence  also  as  a  process 
going  on  within  a  certain  period  of  time.  If  the  period 
assigned  to  the  cause  is  earlier  than,  and  not  simul- 
taneous with,  that  assigned  to  the  effect,  then  the  two 
periods  must  ultimately  be  taken  as  strictly  contiguous : 
that  is,  the  terminal  phase  of  the  cause-process  coin- 
cides in  time  with  the  initial  phase  of  the  effect-process. 


a 


The  line  drawn  above  will  serve  to  represent  the  differ- 
ence between  the  inadequate  and  the  adequate  account 
of  cause  and  effect.   If  an  occurrence-^  be  dated  at  the 


\^' 


76 


CHAPTER  VI 


moment  a,  and  an  occurrence  C  at  the  moment  c,  and 
A  be  then  defined  as  the  cause  of  the   effect  C,  the 
account  is  inadequate;  for  not  only  does  it  involve  the 
above-mentioned  paradox,  that  the  non-existence  of  A 
is  a  condition  for  the  existence  of  C,  but  the  account 
fails  to  assign  any  principle  for  determining  the  interval 
of  time  which  must  elapse  between  the  moment  a  at 
which  A  has  ceased  to  be,  and  the  moment  c  at  which 
C  is  manifested.    If  the  time-interval  between  A  and  C 
is  phenomenally  unfilled,  no  account  can  be  given  of 
its  length ;  we  must  therefore  represent  the  interval  be- 
tween a  and  c  as  occupied  by  a  process  of  change,  say 
from  phase  A  at  moment  a  to  phase  B  at  moment  b, 
and  again  from  this  latter  to  phase  C  at  moment  c. 
We  shall  then  no  longer  speak  of  phase  A  at  moment  a 
as  the  cause  of  phase  C  at  moment  c,  but  rather  of  the 
change  from  A  to  B  within  the  period  of  time  ab  as 
cause  of  the  change  from  B  io  C  within  the  period  of 
time  bc]  where  no  empty  gap  of  time  separates  the 
cause  from  the  effect.    The  cause  in  this  case  may  still 
be  said  to  precede  the  effect,  but  it  is  necessary  to  add 
that  the  temporal   relation  is  one  of  strict  contiguity. 
When  we  can  quantify  the  differences  of  phase,  it  is 
possible  not  only  to  indicate  the  nature  of  the  change 
which  takes  place  within  the  whole  period,  but  to  cor- 
relate the  degree  of  change  from  A  to  B,  and  B  to  C, 
with  the  time-relations  ab  and  be.    In  the  simplest  case, 
the  quantity  or  degree  of  change  is  proportional  to  the 
period  of  time  within  which  the  change  takes  place: 
j  for  instance,  to  illustrate  the  first  law  of  motion.  A,  B,  C 
'  might  stand  for  successively  occupied  positions  of  a 
moving  particle,  so  that  AB  and i9C represent  distances; 


i 

K^ 


V-4- 
S 
<* 


V^^ 


lA  '^ 


V   r 


4 


CAUSE-FACTORS 


77 


then  the  distances  AB,  BC,  would  be  proportional  to 
the  periods  ab,  be.  In  this  way  a  principle  is  supplied 
to  account  for  the  length  of  time  which  must  elapse 
between  the  occurrence  of  the  cause  and  the  occurrence 
of  the  effect,  when  these  occurrences  are  dated  at 
separated  moments  of  time;  and  the  initial  paradox  is 
removed. 


<-^4 


CHAPTER  VII 

THE  CONTINUANT 

§  I.  We  have  found  in  our  analysis  of  the  nature  and 
determination  of  occurrences  that  some  link,  besides 
mere  temporal  and  spatial  connection,  must  exist  between 
one  occurrence  and  another  in  order  that  the  first  may 
be  conceived  as  determinative  of  the  second.  This  link 
I  have  called  the  substantive  continuant,  and  in  this 
chapter  we  shall  examine  the  notion  in  detail,  and  show 
how  it  differs  from  the  traditional  conception  of  sub- 
stance. 

The  simplest  and  most  obvious  illustration  of  the 
continuant  is  the  case  of  the  moving  particle :  thus,  if 
two  movements,  defined  in  character  by  direction  and 
velocity,  and  defined  also  by  reference  to  the  period  of 
time  and  region  of  space  within  which  each  takes  place, 
are  to  be  conceived  as  connected,  in  the  sense  that  the 
character,  date  and  location  of  the  one  is  determinative 
of  the  character,  date  and  location  of  the  other,  then 
such  a  connection  can  only  be  presumed  if  the  same 
material  continuant  is  existentially  manifested  in  the 
two  movements.  Apart  from  the  introduction  of  the 
continuant,  this  simple  example  serves  to  illustrate  the 
way  in  which  identity  and  difference  is  involved  in 
causality.  We  speak  of  two  movements,  and  briefly  call 
the  one  cause  and  the  other  effect.  Inasmuch  as  the 
movements  are  two,  they  cannot  be  identical ;  so  that 
it  may  be  laid  down  as  the  first  and  most  indubitable 


THE  CONTINUANT 


79 


f^  ^ 


^   f 


^   V 


i" 
/ — 


principle  of  causality  that,  whatever  other  subtle  relations 
there  may  be  between  cause  and  effect,  the  relation  of 
non-identity  is  to  be  unequivocally  asserted.    Hence,  | 
before  the  movements  in  question  are  connected  as  cause 
and  effect,  they  must  first  be  distinguished  as  o^Aer  or 
^wo]  and  since  time  and  place  are  the  only  conditions  of 
otherness  which  have  been  conceived  by  the  human 
mind  in  regard  to  physical  phenomena,  the  movements 
in  order  to  be  conceived  as  two,  must  occupy  either 
different  periods  of  time  and  the  same  region  of  space, 
or  different  regions  of  space  and  the  same  period  of 
time,  or  different  periods  of  time  and  different  regions 
of  space.    We  will  suppose  that  the  two  movements, 
connected  as  cause  and  eff"ect,  are  referred  to  different 
periods  of  time  and  to  different  regions  of  space,  and 
proceed  to  consider  their  characterisation  as   regards 
direction  and  velocity.   I  n  the  very  simplest  case  afforded 
by  science  of  causal  relation  between  movements,  the 
direction  and  velocity  of  the  movement  called  cause  is 
identical  with  the  direction  and  velocity  of  the  movement 
called  effect;  in  this  case,  therefore,  cause  and  effect 
are  non-identical  as  regards  temporal  and  spatial  reference, 
but  identical  as  regards  characterisation.  Turning  now 
from  the  adjectival  characterisation  of  the  occurrences 
to  their  substantival  connection,  our  illustration  may  be 
expressed  in  terms  of  the  first  law  of  motion,  as  follows : — 
So  far  as  the  movement  of  a  particle  within  one  period 
of  time  is  causally  determinative   of  its  movements 
within  another  period  of  time,  the  direction  and  velocity 
of  movement  is  the  same  within  these  two  periods. 
Here  the  two  movements  which  are  causally  connected } 
are  movements  of  one  and  the  same  particle ;    so  that  I 


</ 


8o 


CHAPTER  VII 


)^ 


.' 


substantival  identity  is  a  notion  essential  to  the  under- 
standing of  the  formula.  It  is  to  this  substantival 
identity  that  I  refer  when  I  speak  of  a  continuant. 

§  2.  For  logical  purposes  it  replaces  the  term  sub- 
stance, familiar  in  metaphysics;  but  the  various  un- 
founded or  a  priori  characteristics  which  philosophers 
have  attributed  to  substance  must  be  carefully  separated 
from  the  essential  logical  residuum,  and  rejected  from 
the  notion  of  the  continuant.  Thus,  in  the  first  place, 
the  conception  of  continuance  has  been  extended  into 
the  infinite  future  and  the  infinite  past.  In  my  view,  on 
the  other  hand,  the  application  of  continuance  must  be 
strictly  limited  to  the  periods  of  time  in  reference  to 
which  we  can  speak  of  change ;  that  is,  so  far  as  we  are 
justified  in  speaking  of  a  state  or  condition  as  changing 
when  we  pass  in  thought  from  one  period  of  time  to 
another,  so  far  are  we  justified  in  conceiving  of  the 
same  entity  or  continuant  as  preserving  its  existence 
throughout  the  two  given  periods.  This  does  not 
warrant  us  in  asserting  its  existence  either  before  or 
after  these  two  periods.  In  physics,  it  is  true  that 
scientists  have  found  it  convenient  to  postulate  an 
indefinitely  prolonged  existence  into  the  past  and  future 
of  the  ultimate  atoms  which  constitute  matter ;  but  this 
has  no  general  logical  or  philosophical  warrant,  any 
more  than  there  is  philosophical  or  logical  warrant  for 
immortality. 

§  3.  The  next  way  in  which  metaphysicians  have 
characterised  any  continuant  or  substance  in  an  unwar- 
rantable fashion,  is  by  maintaining  that  amid  all  the 
alterations  of  state  or  condition  which  the  substance 
undergoes,  there  are  some  one  or  more  characters  which 


r\ 


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<»•.* 


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-,»^ 


\ 


THE  CONTINUANT 


81 


V 

4 


rf    y 


> 


r-v^  —J 


continue  to  be  manifested  unchanged.    This  position 
ipiay  be  held  in  a  more  or  less  crude  form.     From  my 
Jpoint  of  view,  what  is  important  to  point  out,  is  merely 
I  that  substantival  continuance  does  not  necessarily  imply 
any  adjectival  changelessness.  When  philosophers  like 
Locke  and  Hume  sought  for  significance  in  the  con- 
ception of  identity y  as  substantive  continuance  used  to 
be   called,  they  were  continually  guilty  of  confusing 
continued  existence  of  the  same  substantive  entity  with 
qualitative  or  adjectival  identity  of  character  or  state. 
Failing  to  find  this  qualitative  identity,  Hume  explicitly 
rejected  substantival  identity ;  and  those  who  opposed 
Hume  held  equally  with  him  that  substantive  identity 
could  not  be  maintained  except  in  so  far  as  qualitative 
identity  could  be  established.    I  repeat  then,  that  the 
conception  of  substantive  continuance  does  not  by  itself 
carry  with  it  the  implication  of  unchanged  character 
through  the  period  of  time  to  which  the  substantive 
continuance   applies.     As    in    the    matter   of  absolute 
temporal  permanence,  so  also  in  this  question  of  un- 
changed  character,  the  physicists  have  found  it  convenient 
to  postulate  in  various  forms  an  unchanged  continuance 
of  character  in  the  atoms  or  compound  bodies  which 
constitute  the  matter  of  the  universe.    Though  I  have 
here  called  these  postulates  of  the  physicist  assumptions, 
I  do  not  wish  to  deny  that  some  of  them  may  have 
inductive  warrant;  to  this  we  shall  have  to  return  when 
we  consider  scientific  induction  in  detail. 

§  4.  The  third  and  last  a  priori  attitude  towards  the 
notion  of  the  continuant  must  be  briefly  treated.  This 
is  the  contention  that  the  ultimate  continuant  is  simple 
and  not  compound.    On  this  subject  it  is  perhaps  of 


J  L  ni 


^i 


4 


82 


CHAPTER  VII 


greatest  importance  at  the  present  day  to  distinguish 
between  compound,  in  the  sense  of  a  whole  consisting 
of  parts,  and  compound  in  the  sense  of  involving  inner 
causal  or  dynamic  interaction.  The  former  conception 
raises  no  serious  problem,  the  continued  identity  of  the 
whole  being  obviously  involved  in  the  continued  identi- 
ties of  the  parts.  It  is  possible  however  to  conceive 
of  a  compound  entity  which  continues  to  preserve  its 
identity  through  change  of  time,  although  none  of  the 
parts,  which  appear  from  time  to  time  to  constitute  the 
whole,  can  be  said  to  preserve  their  several  identities. 
This  may  conceivably  be  explained  by  exhibiting  a  law 
or  principle  in  accordance  with  which  the  compound 
continuant  develops  a  changing  character  by  means  of 
the  instrumentality  of  the  dynamic  interactions  amongst 
the  parts  or  components  which  from  time  to  time  consti- 
tute so  to  speak  the  substantival  material  of  which  the 
compound  continuant  is  composed.  Thus  the  law  or  prin- 
ciple according  to  which  the  character  of  the  continuant 
at  one  time  can  be  exhibited  as  depending  upon  its  cha- 
racter at  another  time,  may  be  the  ground  for  asserting 
continued  existential  identity,  although  the  material  com- 
ponents of  this  continuant  are  not  themselves  continuant. 
§  5.  We  began  our  exposition  of  the  continuant  by 
an  illustration  from  physical  science,  showing  how  the 
physical  continuant  is  involved  in  the  simple  formula 
known  as  the  first  law  of  motion.  We  shall  now  bring 
forward  an  illustration  of  approximately  equal  simplicity 
from  the  psychical  sphere.  In  the  physical  illustration 
we  included  reference  to  space  as  well  as  to  time;  in  our 
psychical  illustration  we  shall  drop,  for  the  present,  any 
reference  to  space.   If  a  sensation  characterised  in  some 


i. 


THE  CONTINUANT 


i\ 


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It. 


i'i 


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S3 


way,  and  a  thought  process,  also  characterised  in  some 
way,  occur  one  within  some  period  of  time,  and  the 
other  within  the  same  or  a  different  period  of  time,  then 
the  character  and  date  of  the  sensation  can  only  be 
conceived  as  determinative  of  the  character  and  date  of 
the  thought  process  if,  in  the  simplest  case,  the  same 
psychical  continuant  is  existentially  manifested  both  in 
the  sensation  and  in  the  thought  process.  Precisely  as 
in  the  case  of  physical  phenomena,  change  or  alteration 
in  time  does  not  mean  the  replacement  of  a  sense- 
experience  of  red,  say,  referred  to  one  period  of  time, 
by  a  sense-experience  of  blue  referred  to  another  period 
of  time;  for  the  mere  reference  of  differently  character- 
ised experiences  to  different  periods  of  time  does  not 
constitute  what  we  call  change  or  alteration.  Here  as 
in  the  case  of  the  physical  continuant,  we  can  only 
speak  of  change  or  alteration  by  conceiving  of  an 
existent  which  continues  to  exist  within  both  the  periods 
of  time  to  which  the  change  refers  ;  and  it  is  for  this 
reason  that  we  call  such  an  existent  a  continuant. 

But  the  notion  of  change,  when  applied  to  the  psychical 
continuant,  raises  a  peculiar  problem  when  we  consider 
the  different  kinds  of  experience  referable  to  one  and 
the  same  continuant :  thus  we  may  put  the  question 
whether  it  is  correct  to  speak  of  a  change  of  state  when, 
for  example,  we  refer  a  sensation  to  one  date,  and  a 
thought  or  volition  to  another  date  ;  or  when  we  refer 
say  a  colour  sensation  to  one  date  and  a  sound  sensation 
to  another  date.  The  mere  fact  that  any  colour  sensation 
is  by  definition  different  from  any  sound  sensation,  and 
still  more  that  any  sensation  is  different  from  any  thought 
or  from  any  volition,  does  not  appear  to  justify  us  in 

6 — 2 


i 


84 


CHAPTER  VII 


THE  CONTINUANT 


85 


speaking  of  change  or  alteration  when  such  phases  of 
experience  are  referred  to  different  dates.  .On  the 
other  hand  we  should  with  less  hesitation  speak  of 
change  or  alteration  when  the  differing  experiences 
come  under  the  same  determinable.  A  sensation  of  red 
followed  by  a  sensation  of  blue — blue  and  red  being 
determinates  under  the  same  determinable,  colour — 
would  appear  to  illustrate  the  notion  of  change  of  state 
more  correctly  than  a  sensation  of  red  followed  by  a 
sensation  of  noise,  or  by  a  thought  about  geometrical 
relations,  or  again  by  a  voluntary  decision  to  get  out  of 
bed.  Now  the  real  reason  why  we  apply  the  word  change 
preferably  to  the  first  case  and  not  to  the  second  is 
because  we  suppose  that  the  blue  sensation  has  replaced 
the  red  sensation,  so  that  at  the  time  that  the  blue  is 
manifested,  the  red  has  ceased  to  be  manifested.  It  is  • 
thus  the  cessation  of  one  character  of  our  experience, 
and  its  replacement  by  another,  that  constitutes  the 
essence  of  change. 

§  6.  The  above  analysis  helps  towards  a  solution  of 
the  problem  as  to  what  it  is  that  can  be  said  to  change. 
On  the  one  hand,  it  cannot  be  the  continuant  itself,  nor 
any  of  its  properties,  since  these  are  asserted  to  be 
constant  throughout  the  period  of  time  to  which  the 
process  of  change  is  referred.  Neither  can  it  be  the 
manifestations,  dated  at  time-points,  which  can  be  said 
to  change,  since  these  merely  replace  one  another  from 
instant  to  instant.  The  clue  to  the  problem  is  to  be 
found  in  the  theory  of  the  determinable.  The  character 
of  each  dated  manifestation  is  determinate,  and  a  change 
implies  always  that  the  determinate  character  of  the 
one  manifestation  at  one  instant  is  replaced  at  a  sub- 


-"<'  H 


h 


') 


H 


»■ 


L> 


sequent  instant  by  a  manifestation  having  a  different 
determinate  character  under  the  same  determinable. 
Thus  we  speak  of  temperature  or  colour  or  size  or 
shape,  etc.,  as  changing  or  remaining  constant  during  a 
certain  period  of  time ;  it  is  therefore  the  manifestation 
— not  of  a  determinate — but  of  a  determinable  that  may 
jbe  said  to  change.  But  further,  the  idea  of  change 
involves  not  only  the  adjectival  determinable,  but  also 
the  substantival  determinandum ;  for  change  would 
have  no  meaning  unless  there  were  a  continuant,  which 
was  necessarily  manifested  in  a  mode  characterised  by 
one  or  another  determinate  value  of  a  determinable. 
Thus  the  substantival  determinandum  is  conceived  as 
continually  manifesting  one  or  another  determinate 
character  under  the  same  determinable,  and  being 
potentially  manifestable  in  a  mode  characterised  by 
any  value  of  the  determinable.  This  aspect  of  the 
nature  of  change  leads  to  the  conception  of  that  which 
determines  this  potentiality  to  become  an  actuality ;  in 
other  words  the  conception  of  change  brings  with  it 
the  conception  of  causal  determination. 

To  prevent  one  minor  confusion  it  is  necessary  to 
point  out  that  what  holds  of  change  proper  holds  also  of 
the  continuance  of  the  manifestation  unchanged;  for  the 
fact  of  continuance  as  well  as  of  change  requires  the 
assignment  of  a  cause.  The  fact  that  the  popular  mind 
demands  only  an  explanation  of  change  which  will  assign 
the  event  or  occurrence  operating  as  cause,  is  accounted 
for  by  familiarity  with  unchanged  continuance  in  many 
manifestations.  Actually  the  preceding  unchanged  con- 
tinuance constitutes  in  such  familiar  cases  the  cause  of 
the  subsequent  unchanged  continuance;  but  it  is  only 


»i 


86 


CHAPTER  VII 


THE  CONTINUANT 


87 


when  this  continuance  is  interrupted  that  the  question 
of  the  cause  of  interruption  is  generally  raised.  For  this 
reason  the  conceptions  of  cause  and  of  change  are  always 
supposed  mutually  to  involve  one  another. 

§  7.  The  simple  illustrations  which  we  have  brought 
forward  of  a  physical  continuant  and  a  psychical  con- 
tinuant, have  served  to  introduce  the  view  that  the  two 
notions,  familiarly  known  in  philosophy  as  substance  and 
causality,  are  mutually  dependent  the  one  upon  the  other. 
No  adequate  account  of  causality  can  be  given  without 
reference  to  the  conception  of  substance,  i.e.  of  an  ex- 
istent continuant,  physical  or  psychical;  and  on  the 
other  hand,  we  can  only  assign  properties  to  the  sub- 
stance or  continuant  by  defining  the  modes  according 
to  which  it  is  existentially  manifested  as  a  causal  agent 
or  re-agent.  Thus  what  is  called  a  property  of  a  con- 
tinuant is  not  an  actually  manifested  character,  but  it  de- 
fines what  characters  would  be  phenomenally  manifested 
when  certain  assignable  conditions  occur.  For  example, 
the  elasticity  of  an  extensible  string  illustrates  a  property 
which  we  attribute  to  the  string;  it  defines  in  general 
terms  the  degree  of  length  which  would  be  attained 
were  the  string  exposed  to  a  certain  tensional  force. 
A  property,  therefore,  expresses  a  definable  group  of 
manifestations — not  as  actual — but  as  potential.  The 
general  formula  for  expressing  the  property  of  a  con- 
tinuant c  assumes  the  shape:  if  a  certain  occurrence  de- 
fined as/  were  to  take  place,  in  which  the  continuant  c 
is  patient,  then  a  phenomenal  manifestation  defined  as  q 
would  occur  which  is  determined  by  the  nature  of  c. 

It  should  be  observed  that  continuants — i.e.  in  ordin- 
ary language  things— are  classified  according  to  their 


<  -1 


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properties;  such  familiar  terms  as  solid,  liquid  and 
gaseous,  for  example,  do  not  describe  phenomena  as 
actual  but  as  potential,  and  they  are  typical  of  an  innu- 
merable host  of  terms  in  familiar  use.  It  is  not  altogether 
easy  to  distinguish  those  adjectives  in  common  use  which 
denote  respectively  actualities  and  potentialities  of  mani- 
festation ;  in  fact  the  problem  of  their  distinction  raises 
points  of  philosophical  importance.  As  regSLvds  />/iyszcal 
continuants,  in  predicating  such  adjectives  as  those  of 
position,  shape  or  size,  or  more  generally  of  spatial 
configuration,  I  shall  assume  that  we  are  predicating 
actualities ;  but  these  are  the  only  adjectives  descriptive 
of  merely  physical  phenomena  which  are  regarded  unani- 
mously by  physicists  as  actually  manifested.  I  myself 
hold  that  there  is,  besides  spatial  configuration  and  mo- 
tion, another  physically  actualised  mode,  namely  force  as 
defined  in  statics.  Physicists  appear  to  me  to  maintain 
that,  where  equilibrium  exists,  what  has  been  called 
force  is  merely  an  indication  of  potentiality  for  move- 
ment; so  that  only  the  energy  of  movement  is  actual, 
and  in  static  condition  force  is  held  to  be  a  myth.  But 
in  my  view,  static  force  represents  a  real  condition  of  a 
body;  e.g.  when  a  heavy  body  is  in  equilibrium  on  a  hori- 
zontal surface,  the  force  called  pressure  actually  exists, 
and  is  not  a  mere  measure  of  what  would  take  place  if 
free  motion  were  permitted.  One  evidence  for  this 
view  is  the  recognised  association  of  strain  with  stress  ; 
i.e.  stress  is  a  particular  example  of  force  which  is  cor- 
related with  strain;  this  latter  being  a  geometrical  con- 
ception. The  formula  according  to  which  strain  and 
stress  are  mutually  connected  and  yet  distinguished,  so 
that  they  stand  to  one  another  as  cause  to  effect  or  as 


88 


CHAPTER  VII 


y 


THE  CONTINUANT 


89 


effect  to  cause,  appears  to  me  to  place  them  both  in 
the  category  of  actualities,  since  a  cause  cannot  be  said 
to  be  in  operation  if  we  conceive  it  as  a  mere  potenti- 
ality. I  should  therefore  include  in  what  used  to  be 
called  the  primary  qualities  of  matter,  besides  spatial 
configuration  and  motion,  resistant  force,  this  phrase 
being  preferable,  in  my  opinion,  as  well  as  of  wider  ap- 
plication than  the  dubious  term  'impenetrability.'  My 
definition  of  the  so-called  primary  qualities  is,  therefore, 
that  they  denote  the  adjectives  or  relations  in  terms 
of  which  actual  physical  phenomena  can  be  described ; 
whereas  the  so-called  secondary  qualities  are  properties, 
inasmuch  as  they  denote  potentialities  for  producing 
sensational  effects.  Thus  in  describing  the  colour  of 
the  surface  of  a  body,  we  may  be  defining  something 
physically  actual,  but  we  are  also  most  certainly  defining, 
besides,  what  is  merely  potential ;  viz.,  that  if  a  luminous 
centre,  such  as  the  sun,  is  in  such  spatial  relation  as  to 
radiate  energy  to  the  surface  of  the  body  in  question, 
then  assignable  parts  of  this  energy  will  be  absorbed 
at  the  surface,  and  another  assignable  part  emitted. 
Correlated  with  this  physical  potentiality  of  the  body  is 
a  psychical  potentiality,  which  must  also  be  presented 
partly  in  spatial  terms;  viz.,  that  if  a  living  organism 
susceptible  to  light-impressions  be  in  appropriate  spatial 
relations  to  the  body,  there  will  be  a  visual  sensation 
to  which  the  name  red  primarily  and  properly  applies. 
The  varied  applications  in  physics  of  such  terms  as 
coefficient  or  index  are  obvious  illustrations  of  what, 
from  the  logical  standpoint,  we  regard  as  potentialities 
in  contrast  to  actual  physical  process.  Such  terms  de- 
note what  are  commonly  called   constants;    and  the 


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common  use  of  this  term  will  serve  to  illustrate  the 
notion  of  a  property.    Most  so-called  constants  should 
more  strictly  be  called  'variable  invariables,'  for  only 
a  few  of  them  are  absolutely  invariable;  they  assume 
different  values  for  different  classes  of  bodies,  although 
they  may  remain  potentially  invariable  when  applied  to 
any  one  the  same  body.   This  illustrates  the  point  that 
in  attributing  a  property  to  a  body,  we  imply  that  a 
certain   formula  can   be  asserted  of  the  processes  in 
which  that  body  may  be  concerned,  which  formula  re- 
mains unchanged  on  the  different  occasions  in  which 
the  processes  may  take  place.    It  further  illustrates  the 
point  that  bodies  may  be  classified  according  to  the 
different  values  of  their  determinable    properties,   as 
represented  by  the  different  values  of  the  so-called  con- 
stants.   As  an  example  of  this,  let  us  take  the  adjective 
'ruminant'    So  far  as  we  predicate  this  adjective,  we 
are  certainly  implying  the  existence  of  an  organ  of 
specific  character,  which  could  be  defined  anatomically 
or  structurally  in  terms  of  spatial  arrangement  and, 
say,   mechanical  pressure.    In  addition  to   these  ele- 
ments of  definition,  we  should  further  assume  that  the 
organ  is  persistently  functioning,  though  the  determi- 
nate mode  in  which  it  is  functioning  would  be  changed 
from  time  to  time  in  accordance  with  a  general  formula 
defining  the  potentialities  of  the  organ  to  function  when- 
ever assignable  conditions  may  be  actualised. 

§  8.  So  far  we  have  been  discussing  the  continuant 
chieflyfrom  the  point  of  view  of  deduction;  but  I  propose 
now  to  treat  it  from  the  point  of  view  of  primitive  induc- 
tion, and  to  exhibit  a  constructive  process  by  which,  prior 
to  anything  that  could  be  called  classification  of  organised 


f 


90 


CHAPTER  VII 


THE  CONTINUANT 


91 


or  of  unorganised  bodies,  mankind  have  put  phenomena 
into  groups.  The  symbols  P^,  P^^  P^---  will  represent 
phenomena  characterised  hy  P,  the  suffixes  being  under- 
stood to  indicate  some  kind  of  order — either  temporal  or 
spatial  or  both — which  we  shall  speak  of  as  a  nexus.  The 
first  motive  for  grouping  phenomena  is  the  observation 
of  some  such  nexus :  thus  when  an  order  from  P^  to  P^ 
to  P^toP^,,.  has  been  repeatedly  observed  to  be  main- 
tained under  a  variety  of  circumstances  which,  in  some 
of  the  different  recurrences  have  been  constant  and  in 
others  have  varied,  then  these  phenomena  have  been 
grouped  as  manifestations  of  an  existent  agent.  This 
observed  uniformity  in  the  temporal  succession  of  phe- 
nomena is  then  inferentially  extended  to  apply  to  other 
assumed  phenomena,  regarded  as  modes  inwhich  a  single 
agent  manifests  its  continued  existence.  The  above  con- 
ception of  nexus  therefore  involves  not  only  a  preserved 
temporal  order  of  phenomena,  but  also  reference  of  these 
phenomena  to  a  single  continuant.  In  physical,  as  dis- 
tinguished from  psychical,  phenomena  there  is ,  in  addition 
to  the  temporal  nexus,  a  spatial  nexus,  between  pheno- 
mena presented  in  spatial  contiguity,  representing  modes 
in  which  a  single  material  body  or  occupant  is  spatially 
manifested.  Thus  when  the  spatial  order  of  such  charac- 
ters as  /^i,  /^2J  ^3  •••  is  preserved  throughout  exterior 
processes  which  are  changing  or  remaining  unchanged, 
the  group  of  characters  is  with  special  impressiveness 
taken  to  constitute  a  unity,  and  conceived  as  referable 
to  a  single  occupant  which  maintains  its  form  of  spatial 
nexus,  however  exterior  conditions  may  be  altering. 

The  unity  of  the  occupant  is,  however,  not  stably 
maintained  with  the  same  degree  of  permanence  as 


^>N 


V 


'f 


*-    V( 


f 


^      .A 


^      *. 


\. 


~*     ^1 


is  attached  to  the  temporal  succession  amongst  the 
manifestations  of  the  continuant.  In  this  respect 
spatial  occupants  fall  into  different  classes  according  to 
the  degree  in  which  they  preserve  the  form  of  their 
spatial  nexus :  the  bodies  which  preserve  their  spatial 
nexus  in  the  highest  degree  are  called  solids;  those  that 
preserve  it  in  less  degree  are  called  liquids ;  and  those 
in  which  the  spatial  nexus  can  hardly  be  said  to  be 
preserved  at  all  are  called  gases.  These  degrees  of 
spatial  nexus  actually  depend  upon  the  power  of  bodies 
to  sustain  tension  and  pressure ;  bodies  which  can  sus- 
tain both  are  called  solids;  liquids  cannot  sustain  ten- 
sion, but  only  pressure ;  and  gases,  which  are  also  unable 
to  sustain  tension,  would  lose  their  spatial  nexus  alto- 
gether unless  pressure,  produced  by  external  force,  de- 
termined the  space  which  they  can  be  made  to  occupy. 
Bodies  regarded  as  occupying  spaces  of  defined  shape 
and  size  can  be  divided  into  classes  on  many  different 
fundamenta  divisionis :  for  instance  according  as  they  are 
unorganised  or  organised  in  various  different  forms,  or 
again  according  to  their  chemical  composition.  A  body  is 
classified  according  to  the  mode  in  which  it  maintains 
its  unity  as  a  whole,  this  unity  consisting  in  xkv^  perma- 
nence of  mode  of  presentation  of  its  manifestations, 
while  exterior  conditions  change  or  remain  constant. 
When  the  spatial  unity  which  leads  us  to  conceive  of 
a  body  as  a  single  whole  is  dissolved  by  exterior 
force  which  the  body  is  unable  to  resist,  then  each  of 
the  several  parts  into  which  the  body  becomes  separated 
has  its  own  constitution  as  a  unit;  and  this  constitution 
may  or  may  not  be  generically  the  same  as  that  of  the 
whole  into  which  the  parts  were  previously  united. 


{( 


92 


CHAPTER  VII 


§  9.  The  consideration  of  the  different  degrees  of 
spatial  nexus  exhibited  by  different  spatial  occupants, 
/leads  to  the  notion  of  a  single  continuant-occupant  being 
at  the  same  time  a  system  of  ^^<5-continuants  or  atoms. 
These  sub-continuants  constitute  a  system,  in  the  sense 
that  they  preserve  a  certain  form  of  spatial  nexus  either 
unchanged,  or  else  changing  in  accordance  with  a  causal 
formula  which  expresses  both  the  immanent  causality 
to  be  attributed  to  the  several  sub-continuants,  and  the 
transeunt  causality  to  be  attributed  to  the  interactions 
amongst  these  sub-continuants.  For  example,  the  gas 
that  is  contained  in  a  vessel  is  a  system  of  sub-con- 
tinuants— viz.  the  gaseous  molecules — which  severally 
exhibit  their  own  immanent  causality,  while  their  spa- 
tial nexus,  manifested  as  mutual  pressure,  exhibits  tran- 
seunt causality.  Again  each  molecule  may  be  a  sub- 
system consisting  of  sub-continuants — viz.  atoms — 
which  taken  severally  exhibit  immanent  causality,  and 
taken  in  combination,  transeunt  causality.  Thus,  any 
system  of  sub-continuants  may  be  regarded  either  as  a 
unity  or  whole,  whose  changes  are  determined  under 
the  form  of  immanent  causality;  or  any  such  whole 
may  be  conceived  as  a  system  of  parts  which  are  them- 
selves continuants,  and  whose  processes  are  to  be 
distinguished  from  the  transeunt  processes  involved  in 
their  interactions  in  the  larger  system.  In  future,  the 
phrase  *  single  continuant'  is  to  be  understood  to  include 
a  single  system  of  sub-continuants  or  sub-occupants. 
Such  a  system  may  be  ranged  in  an  order  exhibiting 
higher  and  higher  forms  of  unity;  of  which  the  lowest 
form  is  a  mechanical  system,  and  the  highest  a  psychi- 
cal continuant.     In  a  mere  mechanism,  the  system  as 


■•  << 


•*-Vj 


fr-l 


,4VH. 


♦  * 


i^  f 


^ 


< 

i'.. 


f 


^     ^ 


^  r 


THE  CONTINUANT 


93 


a  whole  can  be  defined  in  terms  of  the  immanent  char- 
acter of  its  parts,  combined  with  the  transeunt  causality 
under  which  the  parts  interact ;  but  the  higher  form  of 
unity  exhibited  in  an  organism  entails  a  much  more 
complex   interrelationship  of  immanent  and  transeunt 
interactions ;  thus  each  part  of  an  organism  undergoes 
processes  which  follow  chemical  and  mechanical  formu- 
lae, which  remain  true  even  when  the  organism  is  dead. 
§  10.    The   forms   of  temporal    and    spatial    nexus 
observed  prior  to  experiment  to  have  been  maintained 
between  groups  of  phenomena,  lead  to  their  constructive 
reference  to  a  single  continuant-occupant.   Primitive  in- 
duction consists  in  the  more  or  less  tacit  inference  that 
where  such  forms  of  nexus  have  been  found  to  obtain 
within  the  range  of  observation,  the  same  form  or  de- 
gree of  nexus  will  be  maintained  in  all  cases  that  have 
not  yet  been  observed.    This  may  be  shortly  expressed 
in  the  proposition:  'All  manifestations  of  a  given  single 
continuant  will  assume  a  specific  form  of  nexus ' ;  and 
the  assignment  of  a  substantive-name,  therefore,  to  any 
manifestation  in  space  and  time,  is  tantamount  to  a  state- 
ment of  uniformity.   1 1  should  be  specially  noted  that  the 
range  of  uniformity  here  asserted  does  not  extend  be- 
yond the  manifestations  of  a  single  individual  continuant. 
Now  this  uniformity  is  not  a  mere  statement  of  the  in- 
variability by  which  we  can  infer  from  certain  pheno- 
mena other  phenomena  in  spatial  and  temporal  connec- 
tion with  these,  but  it  involves  also  a  statement  of  causal 
relation  between  the   phenomena.    The    causality  af- 
firmed is  immanent,  and  is  conceived  as  exhibiting  the 
nature  of  the  individual  continuant  itself.   The  pheno- 
mena in  question  may  be  said  to  co-exist,  the  prefix  co 


94 


CHAPTER  VII 


THE  CONTINUANT 


95 


indicating  that  they  are  manifestations  of  a  single  con- 
tinuant;  for,  as  I  have  elsewhere  pointed  out,  the  phrase 
^co-existence  of  properties  does  not  mean  merely  temporal 
simultaneity  in  the  manifestations,  but  includes  prece- 
dence and  subsequence,  and  also  in  some  cases  spatial 
relations. 

§  1 1.    The  assertion  of  causality  does  not  carry  with  it 
any  implication  of  a  uniformity  of  co-existence,  affirming 
that  the  causal  formula  which  applies  to  the  continuant 
under  consideration  would  hold  of  other  continuants. 
The  nature  of  any  given  individual  is  exhibited  in  a 
uniformity  of  its  manifestations,  which  again,  in  my 
view,  is  a  causal  uniformity.    But  this  species  of  uni- 
formity is  not  one  that  can  be  extended  in  application 
to  the  manifestations  of  other  individuals,  except  so  far 
as  we  have  inductive  evidence  that  the  same  form  of 
immanent  causality  that  expresses  the  nature  of  one 
individual  expresses  also  the  nature  of  other  individuals. 
The  causal  uniformity  v^hxch.  obtains  between  the  mani- 
festations of  a  single  individual  continuant  is  established 
by  more  or  less  direct  and  simple  problematic  methods  ; 
whereas  in  order  to  establish  a  uniformity  of  co-exist- 
ence, the  criteria  of  probability  employed  are  much  more 
exact  and  logically  elaborate.  What  I  have  now  to  main- 
tain is  that  there  are  many  conclusions  reached  by  the 
first  method  which  can  be  said  to  have  been  established 
with  a  very  high  degree  of  probability,  amounting  in 
some  cases  to  practical  certitude.    They  assert  persist- 
ence in  the  form  under  which  the  manifestations  of  a 
single  continuant  may  be  subsumed.    Induction,  on  the 
other  hand,  which  attributes  the  same  form  of  property 
to  one  and  to  another  continuant,  cannot  attain  a  high 


V 


X 


4  V^ 


r  ^ 


-k 

♦ 


1^  > 


degree  of  certitude  by  simple  problematic  methods; 
for  what  enables  us  to  construct  a  class  of  continuants 
so  defined,  is  the  employment  of  more  or  less  scientific 

method. 

§  12.    We  will  now  pass  to  the  consideration  of  the 
relation  of  the  continuant  or  supreme  substantive  to  its 
occurrents  or  modes  of  manifestation.    These  modes  of 
manifestation  have  variously  defined  characters  and  are 
connected  with  one  another  by  temporal  and  spatial 
relations.    Just  as  the  supreme  substantive  is  called  a 
continuant  on  the  ground  that  it  continues  to  be  mani- 
fested throughout  an  indefinitely  prolonged  period  of 
time,  within  which  any  one  manifestation  or  occurrent 
is  referred  to  a  specific  sub-period;  so  it  may  also  be 
called  an  occupant,  on  the  ground  that  its  manifestations 
occupy  an  indefinitely  extended  region  of  space,  within 
which  any  one  manifestation  is  referred  to  a  specific 
sub-region.   The  determinate  sub-period  and  sub-region 
together  define  the  relation  of  one  occurrent  to  others ; 
whereas  the  time  and  space,  of  which  these  sub-periods 
and  sub-regions  are  respectively  parts,  maybe  predicated 
referentially  to  the  continuant-occupant  itself.     Thus 
when  we  say  that  a  continuant  exists  throughout  the 
whole  of  time,  or  an  occupant  throughout  the  whole  of 
space,  we  mean  that  its  several  different  manifestations 
are  to  be  referred  to  one  or  another  of  the  sub-periods 
or  sub-regions  within  this  whole ;  and  if  we  are  concerned 
with  an  animate  or  psychical  continuant,  the  whole  of 
time  or  the  whole  of  space  within  which  it  exists  may 
be  itself  a  sub-period  or  a  sub-region  within  a  larger 
whole ;  since  there  may  be  some  part  of  time  during 
which  it  is  not  manifested  in  any  part  of  space,  and 


96 


CHAPTER  VII 


there  may  be  some  part  of  space  in  which  it  is  not 
manifested  in  any  part  of  time.  This  is  one  of  the  ways 
in  which  predications  (viz.  temporal  and  spatial),  which 
are  primarily  attached  to  existent  manifestations  are 
transferred  to  an  existent  continuant  or  occupant.  We 
are  thus  led  to  enquire  what  other  adjectives  can  be 
predicated  of  a  continuant  which  are  derived  from  the 
characterisations  of  its  several  occurrents  or^  manifesta- 
tions. To  explain  this  we  must  conceive  of  the  unity 
of  the  continuant  as  exhibited  in  causal  formulae  sym- 
bolisable  as  follows : 

P=fJX,     Q=f^'X,    R=/^'X,  

where  each  equation  represents  a  property  of  the  con- 
tinuant. The  small  letters  f  indicate  that  the  property 
or  function  is  defined  determinately,  while  the  capital 
letters  X,  P,  Q,  R,  indicate  that  this  same  determinate 
property  is  exhibited  for  all  determinate  values  which 
Xy  P,  Qy  R  may  assume  at  any  specific  time  or  place. 
Now  as  regards  physical  continuants,  there  are  many 
properties  or  functions  which  exhibit  literally  the  same 
determinate  value  at  any  time  or  in  any  place;  but,  in 
more  complex  organic  continuants,  such  properties  as 
have  been  denoted  by  the  symbol  f  change  from  time 
to  time.  These  temporal  changes  in  the  manifested 
properties  of  things  are  not  irregular,  but  follow  a  law 
(dependent  upon  the  immanent  character  of  the  things) 
which  constitutes  a  property  of  what  may  be  called  a 
higher  order  than  the  properties  symbolised  by /J  which 
directly  define  the  causal  characters  of  actual  manifesta- 
tions. In  the  case  of  complex  organic  continuants, 
therefore,  any  actually  manifested  property  symbolised 
by  a  determinate/* comes  within  a  determinable  /^  which. 


^•^ 


r» 


f' 

U 


i 


\ 

♦ 


THE  CONTINUANT 


97 


7 


I 


V 


in  its  variations,  exhibits  a  determinate  property,  say  i/r, 
j  of  a  higher  order.  The  formula  for  this  higher  order 
of  property  will  depend  upon  the  variable — time — as 
also  upon  the  different  circumstances  which  have 
operated  transeuntly  upon  the  continuant  from  time  to 
time.  The  formula  itself  will,  however,  be  a  formula  of 
immanent  causality,  exhibiting  the  nature  of  the  con- 
tinuant itself  as  determining  the  kind  of  effects  which 
will  be  produced  in  it  under  the  influence  of  external 
circumstances.  It  is  impossible  adequately  to  represent 
this  notion  of  alterable  properties  in  symbols,  but  I 
suggest  the  following  scheme  for  illustrating  the  con- 
ception : 

P  =  xl^p{Xy  VyZ,„,     T^TyT^,..l 
which  degenerates  into  P==/p{Xy  V,  Z,  ..,)  when  the 
7^'s  have  some  determinate  values.    From  this  formula 
the  value  of  P  is  determined  from  the  circumstances 
Xy  Yy  Z,  and  the  times  Tj^^y  Ty,  T^y  at  which  these  cir- 
cumstances have  operated.    In  this  formula  the  capitals 
represent  variables:    of  these  variables,  those  in  the 
bracket  are  independent y  while  the  variable  P  is  dependent 
upon   ttiese  independent    variables,    of  which   it  is  a 
function.    The  small  letter  i/f  represents  a  property  of  the 
higher  order,  having  a  determinate  value  which  is  con- 
stantly manifested  whatever  variations  the  variables  may 
assume  either  actually  or  hypothetically.    By  the  appli- 
cative principle,  we  infer  from  such  a  formula  the  actual 
mode  of  functioning  of  the  continuant,  when  the  deter- 
minables  are  replaced  by  given  determinate  values.   The 
specific  form  of  the  function  i/f,  as  indicating  the  character 
of  the  continuant  itself,  illustrates  immanent  causality; 
but  so  far  as  the  circumstances  Xy  V,  Z  are  due  to  the 

J  L  III  7 


V 


98 


CHAPTER  VII 


operation  of  external  agents,  transeunt  causality  is  in- 
volved in  occasioning  the  actual  values  that  they  may 
assume  from  time  to  time. 

§  13.  From  the  above  attempted  explanation  of  the 
notion  of  property,  it  is  but  a  short  step  to  the  concept  of 
a  continuant;  for  the  main  element  in  the  notion  of  thing 
or  continuant  is  the  permanency  of  functioning  that  can 
be  discerned  in  a  series  of  characterised  manifestations, 
presented  in  the  course  of  time,  as  they  may  be  observed 
in  a  temporally  continuous,  or  discrete,  series  of  acts. 
Thus  the  notion  of  a  continuant  is  constructed  in  terms 
of  temporal  connection  and  causal  determination,  and 
my  particular  views  on  this  subject  may  perhaps  be  best 
explained  by  comparing  my  account  with  Kant's  exposi- 
tion in  the  Critique  of  Pure  Reason,  Kant  holds  that 
there  are  certain  categories,  such  as  substance  and 
causality,  under  which  we  objectify  our  sense-experiences 
in  an  order  of  time ;  whereas  I  prefer  to  treat  substance 
and  causality,  not  as  two  separate  categories,  but  as  two 
aspects  of  a  single  principle  of  construction.  Again,  in- 
stead of  adding  a  third  category — namely  reciprocity — 
to  substance  and  causality,  as  Kant  does,  I  include  reci- 
procity in  my  accountof  immanent  and  transeunt  causality. 
But  such  apparent  differences  between  Kant's  exposition 
and  mine  are  not  important,  and  readers  of  Kant,  by 
putting  together  various  parts  of  his  exposition,  would 
find  at  least  hints  of  all  that  I  have  said.  Thus,  when 
he  says  with  regard  to  his  category  of  substance,  that 
the  idea  of  change  involves  the  idea  of  permanence, 
and  when  this  is  supplemented  by  his  schematism  of 
causality  under  the  form  of  time,  his  view  is  seen  to  be 
in  close  accord  with  my  account  of  the  way  in  which 


\ 


4> 


r 


J 


THE  CONTINUANT 


99 


\ 


r'        ^ 


y 


X 


temporal  causality  and  permanency  of  functioning  enter 
in  the  notion  of  a  continuant ;  although  the  postulate  of 
permanency  refers  in  Kant's  exposition  to  a  *  quantum* 
rather  than  to  the  mode  0/ functioning  yfhxch.  I  attribute 
to  the  continuant.  In  my  view  this  permanency  in  the 
mode  of  functioning  is  inseparable  from  the  property — 
or  form  of  causality — this  form  being  just  that  to  which 
permanence  is  attributed;  whereas  Kant  appears  to 
affirm  that  the  substance  itself  exists  permanently,  and 
that  a  second  permanence  is  to  be  attributed  to  one  of 
its  modes  of  manifestation,  namely  to  its  'quantum.* 
This  postulate  of  his  was,  in  fact,  an  anticipation  of  the 
constancy  of  mass  which  is  a  special  postulate  in  physics ; 
but  no  similar  quantitative  constancy  can  be  attributed 
to  the  higher  substantive  entities,  such  as  the  organism 
or  the  experient.  I  am  inclined  to  attribute  Kant's 
denial  of  the  possibility  of  rationalising  psychology  to 
his  rather  exclusive  consideration  of  the  forms  in  which 
the  principles  of  physics  can  be  generalised  and  formu- 
lated in  precise  mathematical  conceptions.  Thus  my 
account  presents  a  more  general  conception  of  substan- 
tive continuance,  which  applies  equally  to  the  notion  of 
a  conscious  experient  on  the  one  hand,  and  to  a  hypo- 
thetical physical  atom  on  the  other.  The  unity  which  I 
ascribe  to  the  continuant  is  a  causal  unity  of  connection 
between  its  temporally  or  spatially  separated  manifesta- 
tions; an  observed  or  assumed  causal  formula,  under 
which  the  character  of  these  manifestations  may  be 
subsumed,  is  the  sole  ground  for  regarding  them  as 
manifestations  of  one  and  the  same  continuant.  I  have 
also  attempted  to  render  clear  the  difficult  conception 
of  the  union  of  permanence  with  change.    It  is  natural 

7—2 


100 


CHAPTER  VII 


ii 


THE  CONTINUANT 


lOI 


to  ascribe  change  to  the  modes  of  manifestation,  and 
permanence  to  the  substance  to  which  these  manifesta- 
tions are  referred;  but  this  is  an  inadequate  expression 
of  the  antithesis;  for,  to  express  the  matter  accurately, 
the  only  things  which  can  be  said  temporally  to  exist 
are  the  manifestations  themselves ;  thus  our  first  defini- 
tion of  the  continuant  is  that  it  is  merely  the  sum  of  all 
the  manifestations.  This  of  course  does  not  mean  that 
manifestations  of  reality  are  taken  indiscriminately, 
mentally  added  together,  and  their  sum  called  a  con- 
tinuant; what  is  meant  is  that  certain  manifestations  of 
reality,  between  which  a  unique  kind  of  relation  can  be 
predicated,  together  constitute  a  genuine  whole  or  unity, 
to  which  the  name  continuant  may  be  given.  This  type 
of  relation,  which  constitutes  the  unity  of  a  single  con- 
tinuant, is  conceived  primarily  as  one  of  immanent 
causality,  while  it  is  transeunt  causality  that  constitutes 
the  ground  for  asserting  a  plurality  of  non-identical 
continuants  whose  manifestations  can  be  said  to  belong 
to  one  universe  of  reality. 

§  14.  All  the  conceptions  expounded  in  this  chapter 
are  virtually  denied  by  a  school  of  philosophers  to-day. 
In  particular  they  regard  the  conception  of  change  as 
fictitious,  and  substitute  for  it  merely  differently  charac- 
terised phenomena  referred  to  non-identical  dates. 
Whenever  there  is  a  spatio-temporal  nexus  between 
phenomena,  the  locating  and  dating  of  the  occurrents 
is  such  that  these  may  be  conceived  as  a  whole.  Such 
a  whole  is  of  the  kind  which  we  have  described  as  ex- 
tensional',  and  so  far  as  extensional  wholes  are  admitted 
by  the  scientist,  no  more  transcendental  conception  than 

^  Part  II,  Chapter  VII,  §  8.  ^ 


•r     * 


♦V 


*  1 


4^ 


il 


\ 


\ 


that  of  a  whole  constituted  by  the  binding  relations  of 
time  and  space  is  required ;  and  hence  the  philosophers 
who  reject  the  conception  of  a  continuant  are  satisfied 
to  replace  it  by  the  notion  of  such  an  extensional  whole. 
But  the  stability  of  a  spatio-temporal  nexus  cannot,  I 
maintain,  be  explained  without  the  conception  of  a 
continuant,  which,  in  my  view,  is  a  priori  va  the  Kantian 
sense,  and  not  derived  from  the  analysis  of  experimental 
data.  Given  the  conception,  however,  it  is  a  question 
of  mere  experience  to  what  set  of  phenomena  the  a 
priori  notion  is  to  be  applied.  In  attempting  to  avoid 
this  conception,  it  appears  to  me  that  my  opponents 
alternate  between  a  purely  physical  and  a  supposititious 
perceptual  account  of  the  facts.  Thus  in  one  breath 
they  shelve  the  physical  continuant  by  supposing  that 
the  percipient  is  observing  a  continuity  in  the  qualitative 

I  changes  of  the  object  perceived;  and  while  in  this  way 
rejecting  any  physical  continuant,  they  have  recourse  to 
I  a  psychical  continuant — namely  the  percipient.  Here 
I  submit  that  the  perception  by  any  individual  of  certain 
processes  offers  no  explanation  whatever  of  what  in 
objective  reality  determines  the  stability  of  any  given 
nexus.  Then  again,  on  the  other  hand,  when  it  is  urged 
that  the  upholders  of  this  view  are  all  along  assuming 
a  psychical  continuant — viz.  the  percipient — which  from 
their  standpoint  must  be  repudiated,  they,  in  effect, 
retort  that  it  is  quite  unnecessary  to  postulate  any 
psychical  continuant,  inasmuch  as  the  nervous  system 
itself  will  take  the  place  of  the  ordinary  conception  of 
an  ego.  Here  then  they  only  eliminate  the  psychical 
continuant  by  reinstating  the  physical  continuant. 


) 


CHAPTER  VIII 

APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND 

§  I .  The  science  of  psychology — so  far  as  it  is  purified 
from  all  reference  to  the  physical — uses  the  conception 
of  immanent  causality  within  the  systematised  unity 
constituted  by  a  single  individual  experient.  In  other 
words,  pure  psychology  abstracts  from  any  transeunt 
causality  which  may  be  actually  operative  in  the  inter- 
actions between  an  experient  and  the  material  world,  or 
between  one  and  another  experient.  The  different 
phases  within  the  experience  of  an  individual  are  con- 
ceived as  related  temporally  and  not  spatially  ;  hence 
the  form  of  space  under  which  we  conceive  transeunt 
causality  in  the  physical  universe,  does  not  apply  within 
the  individual's  experience.  A  modified  form  of  transeunt 
causality  is  applicable  however  to  the  interactions 
amongst  the  distinguishable  phases  revealed  by  a  funda- 
mental analysis  of  conscious  experience.  If,  for  instance, 
we  distinguished  between  merely  sensational  processes, 
on  the  one  hand,  and  active  or  purposive  processes  on 
the  other,  we  might  establish  some  kind  of  uniformity 
which  would  determine  the  course  of  a  sensational 
process  so  far  as  it  was  uninfluenced  by  active  purpose. 
Again  a  process  of  deliberation  might  be  known  to 
pursue  a  course  of  its  own  independently  of  the 
changes  occurring  in  the  sensational  process.  In  this 
way  a  relative  and  partial  independence  might  be 
attributed  to  the  sensational  and  deliberative  processes 


*'T 


^>- 


*■  i' 


'  ii 


s» 


j 


11 


L]»» 


^0^ 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     103 

respectively,  each  proceeding  according  to  its  own  law. 
But,  if  we  suppose  that  at  certain  periods  the  deliberative 
process  exerts  a  determinative  influence  upon  the  further 
course  of  the  sensations,  this  influence  is  analogous  to 
transeunt  causality,  with  the  difference  that  the  causality 
is  not  conceived  as  passing  across  from  one  to  another 
substantive  existent,  but  only  from  one  to  another  phase 
of  experience  within  the  unity  of  a  single  substantive 
existent.  Another  rough  illustration  of  a  similar  type 
may  be  taken  from  associative  as  distinguished  from 
attentive  processes.  Thus  the  forms  assumed  in  the 
flow  of  images  or  ideas  may  be  supposed  to  depend 
essentially  in  the  first  instance  upon  the  time  order  of 
past  experiences,  and  the  frequency  or  recency  of  these 
experiences.  In  so  far  as  this  is  so,  the  course  of 
thought  has  a  law  of  its  own  which  operates  independ- 
ently of  purposive  control.  But  at  such  periods,  when 
the  felt  interest  of  the  thinking  subject  modifies  the 
course  of  associations,  and  determines  the  flow  of  images 
or  of  ideas  to  be  other  than  they  would  have  been  as 
the  result  of  mere  association,  a  form  of  mental  causality 
is  operative  which  is  more  or  less  analogous  to  transeunt 
causality.  Those  psychologists  who  explicitly  attribute 
activity  to  the  subject  may  be  said  to  use  the  conception 
of  transeunt  causality  in  an  even  more  literal  sense  than 
that  which  I  have  so  far  suggested  :  for  they  hold  that, 
apart  from  subjective  activity,  mental  processes  would 
pursue  a  course  determined  on  principles  quasi-me- 
chanical, these  quasi-mechanical  processes,  constituting 
a  sort  of  non-ego.  Over  against  these,  the  subject  or 
true  ego  is  conceived  as  an  agent  having  the  power, 
which  it  exercises  from  time  to  time,  of  controlling  or 


^V- 


I 


104 


CHAPTER  VIII 


modifying  the  processes  which  apart  from  such  activity 
would  proceed  purely  mechanically.  This  splitting  up  of 
experience  into  mechanism  and  active  control  tends,  in 
my  view,  to  misrepresent  the  case,  if  it  leads  to  a  con- 
ception of  the  subject  as  purely  abstract.  The  subject 
as  active  must  be  conceived  as  a  determinate  phase  of 
experience,  which  stands  from  time  to  time  in  definable 
and  alterable  relations  to  the  processes  that  may  be  said 
to  be  actively  controlled.  The  two  most  fundamental 
of  these  relations  are  called  respectively  feeling  and 
cognition ;  according  to  the  mode  in  which  experiences 
arouse  feeling,  and  according  to  the  manner  or  extent 
in  which  they  are  cognised,  so  is  the  exercise  of  con- 
trolling activity  determined.  Thus  feeling  and  cognition 
operate  as  psychical  forces  which  are  analogous  to 
physical  forces,  except  that  the  latter  involve  spatial 
relations. 

§  2.  Another  quite  unambiguous  example  of  transeunt 
causality  is  the  action  of  the  psychical  on  the  physical, 
and  the  apparently  simultaneous  action  of  the  physical 
on  the  psychical.  We  may  venture  to  speculate  that 
various  phases  of  psychical  process  which  proceed  con- 
temporaneously with  physiological  (and  in  particular 
neural)  processes,  can  be  described  in  terms  of  the 
same  number  of  distinct  determinables  as  the  neural 
processes.  It  is  therefore  theoretically  possible  to 
predicate  of  such  phases  of  mentality  a  one-one 
correspondence  with  the  neural  processes,  where  the 
term  **  one-one  correspondence  "  is  understood  as 
equivalent  to  reciprocal  inferability.  This  means  that 
when  (if  ever)  psycho-physical  knowledge  has  been 
adequately  advanced,  it  will  be  possible  from  a  knowledge 


<L 


i 


.#i 


y 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     105 

of  the  character  of  any  neural  process  that  may  have 
occurred,  to  infer  the  character  of  certain  of  the  con- 
temporaneously occurring  phases  of  mentality,  and 
conversely.  This  will  give  a  restricted  validity  to  the 
conception  of  neutral  psycho-physiological  parallelism, 
the  word  *' neutral"  indicating  that  the  theory  does  not 
prejudge  the  question  whether  the  assertion  of  corre- 
spondence implies  an  assertion  of  causal  connection  or 
not.  The  conception  is  restricted  in  the  sense  that, 
besides  the  phases  of  mentality  which  correspond  in 
their  changes  to  the  changes  of  neural  process,  there 
are  other  psychical  phases  to  which  no  changes  of 
neural  process  correspond.  If  the  former  be  provision- 
ally called  sensational  experiences,  I  hold  that  these  are 
to  be  clearly  distinguished  from  cognition  and  feeling, 
which  constitute  the  fundamental  aspects  of  psychical 
process  to  which  no  neural  processes  correspond,  and 
which  may  be  provisionally  defined  as  variable  or 
alterable  relations  or  attitudes  towards  the  sense- 
experiences.  This  may  be  otherwise  expressed  in  the 
assertion  that  there  is  no  direct  correspondence  between 
the  phases  of  cognition  and  feeling  on  the  one  side,  and 
changes  of  neural  process  on  the  other  side.  For,  if  we 
assume  that  the  changes  of  sensation  are  caused  by 
changes  of  neural  process,  it  follows  that,  in  so  far  as 
these  changes  of  sensation  determine  changes  in  the 
phases  of  feeling  and  cognition,  there  will  be  indirectly 
a  correspondence  between  the  latter  and  the  neural 
processes.  The  more  rigidly  we  insist  that  the  phases 
of  cognition  and  of  feeling  are  occasioned  in  their 
changes  by  the  changes  in  the  sense-experiences  to 
which  they  attach,  the  more  clearly  shall  we  realise  the 


io6 


CHAPTER  VIII 


fundamental  distinction  between  the  direct  corre- 
spondence of  the  sensations  to  the  neural  processes, 
and  the  indirect  correspondence  of  these  latter  to  the 
phases  of  cognition  and  feeling.  If,  for  instance,  the 
changes  of  neural  process  are  regarded  as  the  sufficient 
causal  determinants  of  the  changes  in  the  sense-experi- 
ence, and  these  changes  of  sensation  as  sufficiently 
accounting  for  the  cognition  and  feeling  attaching  to 
them,  then  there  can  be  no  remaining  modes  corre- 
sponding to  the  phases  of  active  cognising  and  feeling 
in  which  the  neural  processes  could  be  described  as 
changeable.  It  appears  to  me  that  all  the  experimental 
work,  which  endeavours  to  establish  laws  connecting 
feeling,  for  example,  with  sensational  or  physiological 
changes,  adopts  precisely  this  same  hypothesis.  In 
these  experiments  an  attempt  is  made  to  formulate  in 
general  terms  the  sort  of  character  which  the  physio- 
logical processes  must  have,  in  order  to  account  for  the 
accompanying  feeling  as  being  more  or  less  pleasurable 
or  painful.  It  is  assumed  in  these  experiments  that  all 
the  changeable  modes  of  neural  processes  have  as  their 
correspondents  changeable  modes  of  sensational  ex- 
periences. There  is  never  any  hint  that  the  physiological 
processes  could  be  changed  also  in  some  further  mode 
corresponding  to  the  changeable  phases  of  feeling. 
Similarly  in  the  case  of  cognition,  where  by  this  term  is 
meant — not  merely  awareness  of  a  sensation  but — the 
cognising  it  as  having  a  certain  character.  Thus,  when 
I  speak  of  changes  in  the  cognitive  phase,  I  mean  to 
refer  to  such  changes  as  apprehending  one  sensation  as 
red,  and  apprehending  another  as  blue.  In  this  case 
again,  physiologists  implicitly  assume  that  all  the  modes 


f 


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1 


/ 


-< 


^\'f 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     107 

in  which  the  neural  processes  can  be  described  as 
changeable  are  exhausted  in  accounting  for  the  sensa- 
tions being  red  or  blue.  There  is  therefore  no  residual 
mode  according  to  which  these  neural  processes  could 
be  supposed  to  vary  or  change  which  might  correspond 
to  the  psychical  fact  of  apprehending  the  sensation  as 
being  red  or  as  being  blue.  A  closer  examination  of  the 
nature  of  cognition  and  feeling  will  further  confirm  this 
view ;  for,  while  the  sensational  processes  may  be  sup- 
posed to  be  entirely  accounted  for  by  the  actual  neural 
processes  that  are  contemporaneously  occurring,  the 
cognition  and  feeling  which  attach  to  these  sensational 
processes  are  partly  determined  by  past  and  possibly 
remote  experiences,  and  therefore  cannot  be  wholly 
accounted  for  by  the  present  sensational  and  contem- 
poraneous neural  processes. 

§  3.  A  serious  objection  might  be  taken  to  the  above 
account,  on  the  ground  that  I  have  omitted  any  refer- 
ence to  those  more  central  cerebral  processes  which 
physiologists  describe  as  underlying  the  active  phases 
of  cognition.  In  this  connection  I  will  introduce  the 
word  effort  or  strain;  and  I  wish  to  suggest  that  the 
phenomenon  of  effort  constitutes  a  link  between  the 
physio-sensational  mechanism  on  the  one  hand,  and  the 
subjective  control  which  is  exerted  upon  this  mechanism 
on  the  other  hand.  I  maintain  that  the  phenomenon  of 
'  effort  or  strain  has  both  a  physiological  and  a  sensa- 
tional aspect:  that  is  to  say,  it  involves  a  process  in  the 
physiological  mechanism  which,  like  other  physiological 
processes,  entails  a  corresponding  sensation ;  it  is  there- 
fore legitimate  to  speak  of  an  effort-sensation,  and  the 
term  indicates  that  there  are  certain  analogies  between 


K 


io8 


CHAPTER  VIII 


this  kind  of  modification  of  conscious  experience  and 
other  sensations  such  as  visual  and  auditory.  Thus, 
there  may  be  differences  of  intensity  and  even  of  quality 
in  the  experiences  of  effort,  correlated  with  differences 
of  intensity  and  of  locality  in  the  underlying  processes 
of  the  neural  mechanism.  Again,  an  effort-sensation 
resembles  other  sensations  in  the  further  respect  that 
at  any  moment  it  may  be  more  or  less  determinately 
characterised  in  an  attitude  of  cognition ;  and  this  act 
of  characterisation  may  reach  such  a  limit  of  indeter- 
minateness  that  we  should  say  in  ordinary  language 
that  the  sensation  was  not  being  cognised  at  all.  Lastly, 
a  character  may  be  attributed  to  these  sensations  apart 
from  any  cognition  of  it  by  the  experient ;  that  is  to  say, 
this  character  is  unchanged  by  his  more  or  less  deter- 
minate cognition  of  it.  In  all  these  respects,  effort- 
sensations  resemble  what  are  ordinarily  known  as 
sensations;  there  are  two  important  points,  however,  in 
which  they  differ.  Briefly  these  are  (i)  that  they  are 
subjectively  initiated  ;  and  (2)  that  they  entail  directly 
changes  in  the  neural  processes  which  indirectly  produce 
effects  intended  by  the  subject — the  term  'intention' 
implying  foreknowledge.  It  is  important  to  point  out 
that  the  subject  does  not  of  course  know  what  sort  of 
neural  processes  are  taking  place,  although  to  him  is  to 
be  attributed  the  initiation  of  these  processes.  The 
sense  in  which  his  activity  is  guided  by  knowledge  is 
expressed  by  his  foreknowledge  of  the  effects  upon  his 
future  sensations  or  perceptions  which  will  be  causally 
determined  by  the  particular  operation  that  he  initiates 
in  the  neural  processes.  Thus  we  attribute  to  subjective 
initiation  various  physical  effects  which,  for  our  present 


1 

> 
f 


<■■ 


^> 


w 


p 
f 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     109 

purpose,  may  be  roughly  divided  into  two  classes,  de- 
scribable  as  inner  and  outer  respectively.  The  inner 
physical  effects  are  those  which  are  unknown,  the  outer 
are  those  which  3.re/oreknown.  As  physical  occurrences, 
the  inner  are  causally  determinative  of  the  outer  ;  but, 
in  our  analysis  of  the  mental  processes  involved,  we 
have  to  maintain,  what  may  appear  paradoxical,  that  it 
is  \ki^  foreknowledge  of  the  outer  which  causally  deter- 
mines the  occurrence  of  the  inner.  One,  and  perhaps 
the  most  fundamental  aspect  of  this  foreknowledge  is 
when  the  outer  effects  are  imaginatively  characterised  by 
the  subject  in  their  spatial  relations.  The  terms  inner 
and  outer  may  be  taken  almost  literally  as  defining  what 
has  been  going  on  inside  the  organism,  as  distinguished 
from  what  will  be  going  on  outside  the  organism.  When 
the  intention  of  the  subject  has  been  actualised,  what 
he  perceives  is  presented  as  (so  to  speak)  external  in 
its  spatial  relations,  and  it  is  these  externally  mani- 
fested effects  which  are  foreknown  or  prospectively 
imaged. 

§  4.  We  have  said  that  the  operation  upon  the  neural 
mechanism,  which  appears  to  involve  strain  or  effort,  is 
initiated  by  the  subject.  In  using  this  phrase  I  am  not 
thinking  of  the  subject  in  a  merely  abstract  sense,  or  as 
a  sort  of  transcendental  ego  (to  use  Kant's  phrase) ;  but, 
on  the  contrary,  of  a  process  precisely  definable  in  terms 
of  mental  phase.  In  fact,  as  I  have  already  attempted 
to  show,  it  is  a  mode  of  feeling  and  a  form  of  cognition 
which  jointly  determine  the  specific  operation  upon  the 
mechanism.  Under  the  various  modes  of  feeling  I  wish 
to  include  not  only  the  hedonic  variations  of  pleasure 
and  pain,  according  to  which  experiences  are  felt  as 


\ 


no 


CHAPTER  VIII 


more  or  less  pleasant  or  unpleasant,  but  also  the  modes 
of  conation,  according  to  which  we  feel  more  or  less 
strongly  attracted  or  repelled  by  different  experiences. 
Again  cognition  should  be  understood  to  include  not 
only  the  foreknowledge  of  the  finally  intended  effects, 
but  also  (as  higher  forms  of  knowledge  develop)  the 
foreknowledge  of  the  external  means  which  must  be 
employed  to  produce  these  final  effects ;  the  special  term 
purposive  is  used  to  describe  voluntary  action  of  this 
higher  kind.  From  the  above  analysis  it  is  clear  then, 
that  in  the  psychical  determination  of  physical  effects, 
foreknowledge  is  involved,  and  we  have  attributed  to 
foreknowledge  real  causal  efficiency.  If  this  foreknow- 
ledge could  be  reduced  to  merely  physical  or  physio- 
logical terms,  we  should  have  to  regard  mental  causality 
as  an  illusion;  and  those  psychologists  who  hold  that 
the  changing  phases  of  cognition  are  represented  by 
corresponding  neural  processes,  do  in  effect  deny  any 
genuine  validity  to  the  conception  of  mental  causality. 
§  5.  Closer  examination  of  subjective  activity  intro- 
duces the  notion  of  attention ;  for  I  hold  that  the  most 
important  consideration  in  any  account  of  cognition  is 
the  different  degrees  of  determinateness  with  which  the 
character  of  an  object  may  be  apprehended  in  an  atti- 
tude of  attention.  It  is  implicitly  maintained  by  some 
psychologists  that  a  sensational  process  can  only 
properly  be  said  to  occur  when  the  subject  is  cognising 
the  sensation ;  so  that  where  there  is  no  such  cognition, 
these  psychologists  maintain  that  there  is  no  sensation; 
and  all  that  could  be  asserted  (as  corresponding  to  the 
non-occurrent  sensation)  would  be  a  neural  process  which 
could  be  defined  by  the  physiologist.    Now,  whether 


1 


VM 


1 


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*.■  , 


1 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     in 

this  contention  is  sound  or  not,  it  must  be  agreed  by  all 
that,  when  the  character  of  a  sense-experience  is  the 
object  of  cognition,  it  may  be  cognised  by  the  subject 
with  very  different  degrees  of  determinateness  or  in- 
determinateness.  It  will  also  be  almost  unanimously 
agreed  that  the  function  of  attention  is  to  render  the 
object  attended  to  more  determinately  cognised;  and 
that  continued  attention  to  one  and  the  same  object  does, 
in  effect,  produce  this  result.  Increasing  determinate- 
ness of  cognition  might  thus  be  marked  off  as  the  e^ec^ 
of  attention.  When  further  we  say  that  attention  in- 
volves activity,  and  attribute  this  activity  to  the  subject 
\  itself,  we  are  attributing  the  cause  of  the  process  to  the 
agency  of  the  subject.  But  this  does  not  explain  why 
a  more  determinate  knowledge  of  one  object  rather 
than  of  another  is  being  developed ;  and  to  account  for 
this  a  defined  purpose  is  to  be  attributed  to  the  subject, 
the  achievement  of  which  demands  this  further  de- 
terminate cognition  as  means.  It  would  be  artificially 
formal,  however,  to  draw  an  absolute  line  of  distinction 
between  means  and  end ;  our  attention  may,  for  instance, 
be  momentarily  diverted  to  an  intrusive  sense-experience, 
in  which  case  the  more  determinate  characterisation  of 
the  intruder  may  be  an  end  in  itself  of  the  momentarily 
diverted  attention,  and  not,  at  the  time,  pursued  in  the 
achievement  of  any  more  remote  end.  Incidentally  it 
may  be  observed  that  if  the  sense-experience  still  con- 
tinues, when  attention  has  ceased  to  be  diverted  to  it, 
it  may  operate  as  a  disturbing  factor  in  the  attention 
which  reverts  to  its  previous  object. 

We  are  now  in  a  position  to  distinguish  two  types  of 
subjective  activity,    (i)  The  operation  of  the  subject 


112 


CHAPTER  VIII 


upon  the  sensational  ^roc^sses,  and  directly  or  indirectly 
upon  the  external  physical  effects  presented  to  percep- 
\  tion,  discussed  in  our  previous  general  account,   may 
I  conveniently  be  called  motor,  because  its  actual  mani- 
festation in  the  physical  world  involves  a  change  or 
maintenance  of  spatial  position.     This  motor  activity 
which  produces  or  prevents  overt  physical  movement  is 
to  be  distinguished  from  (2)  the  activity  of  attention, 
which  appears  to  involve  only  a  furthering  or  develop- 
ment in  the  determinateness  of  cognition.     Either  of 
these  forms  of  activity  may  entail  a  more  or  less  intense 
effort :  corresponding  to  the  first  we  may  speak  of  motor 
effort,  and  corresponding  to  the  second  attentive  effort; 
and  both  motor  effort  and  attentive  effort  may  have  a 
quasi-sensational  as  well  as  a  neural  or  physiological  side. 
In  actual  and  overt  movement  we  can  trace  the  neural 
processes  which  entail  corresponding  sensations ;  but  in 
the  case  of  mere  attention,  where  there  is  no  overt  move- 
ment, we  have  to  consider  what  may  be  the  nature  of  the 
physiological  processes  which  underlie  the  effort-experi- 
ence that  seems  often  to  accompany  attentive  processes. 
In  motor  activity  we  assumed  an  operation  upon  the 
neural  processes  underlying  sensation]  in  a  similar  way, 
we  may  perhaps  assume  that  the  activity  of  inner  attention 
entails  an  operation  upon  the  neural  processes  underlying 
imagery.   If  this  hypothesis  is  to  be  consistently  applied, 
our  account  of  ** restricted  parallelism"  must  be  extended 
to  include  a  strict  correspondence — not  only  between 
sensational    experiences   and  their  neural  accompani- 
ments— but  also  between  image  experiences  and  their 
neural  accompaniments.     This  may  be  done  without 
prejudging  the  physiological  question,  irrelevant  to  our 


><> 


*•» 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     113 

present  purpose,  whether,  underlying  any  sense-impres- 
sion and  that  sense-image  which  is  a  sort  of  copy  of  it 
is  the  same  neural  process  in  a  different  form,  or  locally 
distinct  neural  processes.  The  suggestion  I  wish  to  put 
forward  is  that  such  effort  as  appears  to  be  experienced 
in  inner  thinking  is  due  to  the  occurrence  of  imagery 
entailed  in  operating  upon  the  neural  processes.  With- 
out discussing  this  problem,  I  may  point  out  a  possible 
confusion  between  what  is  properly  to  be  called  the 
effort  involved  in  thinking,  and  the  difficulty  of  such 
thinking.  For  a  given  subject,  the  difficulty  of  making 
further  progress  in  thinking  on  a  certain  topic — so  far 
from  implying  intensification  of  effort — may  lead  him 
to  cease  thinking  further  on  the  subject.  This  leaves 
the  problem  still  to  be  examined  whether,  so  long  as 
there  is  a  continuance  of  the  thinking  process,  more  or 
less  effort  is  involved.  Evidence  of  such  effort  may  not 
easily  be  found  in  direct  introspection,  and  may  have  to 
be  sought  indirectly  in  such  effects  as  a  diminution  in 
physiological  or  intellectual  energy;  i.e.  in  future  power 
of  doing  work. 

§  6.  I  will  now  give,  more  explicitly,  my  grounds  for 
the  hypothesis  that  the  changes  of  cognitive  phase  have 
not  counterparts  in  the  changes  of  neural  process ;  con- 
sidering in  succession  the  three  main  features  charac- 
teristic of  even  the  simplest  cognitive  process. 

(i )  We  may  agree  that  perception  denotes  essentially 
the  mode  of  cognition  in  which  objects  are  apprehended 
in  their  temporal  or  spatial  connections  or  relations. 
We  may  further  agree  that  corresponding  to  the  rela- 
tions of  time  which  we  predicate  of  objects  cognised, 
there  are  actual  relations  of  time  subsisting  between  the 

J  L  ni  8 


i/\ 


/ 


114 


CHAPTER  VIII 


neural  occurrences  which  underlie  the  experiences  whose 
temporal  relations  are  apprehended  in  perception.  The 
dating  and  temporal  measuring  of  these  neural  processes 
by  the  physiologist  would  exhaust  what  could  be  stated 
about  them,  when  their  mode  or  locality  had  been  de- 
fined. There  appears  therefore  to  be  no  discoverable 
further  mode  of  neural  process  which  should  correspond 
to  the  mental  act  of  cognitively  defining  the  temporal 
relation.  What  holds  of  time,  holds  of  space :  the  spatial 
relations  between  the  several  areas,  of  which  the  stimu- 
lation causes  sensations,  may  be  assumed  to  vary  with 
the  spatial  relations  as  apprehended  in  the  object 
perceived,  in  accordance  with  some  formula.  And  the  . 
physiologist  may  also  correlate  the  locality  of  the  stimu- 
lations of  the  end-organs  with  the  modes  of  neurosis 
or  the  cerebral  localities  of  the  physiologically  central 
processes.  This  again  would  exhaust  his  account  of 
the  physiological  processes,  and  there  would  be  no  other 
mode  of  variation  which  could  be  correlated  with  the 
act  of  cognitive  spatialisation. 

(2)  Consider  next  the  feature  of  elementary  cognition 
which  is  involved  in  the  act  of  comparison.  In  the 
simplest  case  this  act  is  a  cognitive  determination  of  a 
relation  of  difference  or  of  agreement.  We  may  assume 
that  the  respects  in  which  our  sense  experiences  agree 
or  differ  correspond  to  the  respects  in  which  the  under- 
lying neural  processes  agree  or  differ.  But  as  in  the 
case  of  spatial  and  temporal  connecting,  there  does  not 
appear  to  be  any  further  mode  in  which  the  neural  pro- 
cesses could  vary  which  should  correspond  to  the  act 
of  apprehending  such  relations  of  agreement  or  dif- 
ference ;  these  might  or  might  not  take  place  without 


♦  I 


f* 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     115 

affecting  the  agreements  or  differences  themselves. 
This  same  argument  applies  even  if  those  theorists  are 
right  who  maintain  that  the  only  way  in  which  we  cog- 
nise the  characters  of  our  experience  is  by  cognising 
relations  of  agreement  or  of  difference  between  one 
experience  and  another;  for  here,  as  in  the  previous 
case,  there  can  be  no  physiological  correlate  for  cognition 
in  general.  If  we  pass  for  a  moment  to  the  higher  forms 
of  cognition  which  constitute  the  special  province  of 
logic,  such  relations,  for  instance,  as  those  of  identity, 
of  substantive  to  adjective,  or  of  cause  to  effect,  it  is 
still  more  impossible  to  conceive  of  modes  of  variation 
of  neural  process  which  could  be  correlated  with  the 
occurrence  of  such  acts  of  conceptual  cognition. 

(3)  The  third  feature  of  the  cognitive  process  to  which 
I  wish  to  draw  special  attention  is  the  variability  of  the 
degrees  of  determinateness  or  indeterminateness  with 
which  an  experience  may  be  cognitively  characterised. 
This  feature  of  cognition  may  be  said  to  include  the 
features  treated  under  the  two  previous  heads,  and  to 
constitute  the  most  fundamental  ground  for  our  denial 
of  a  physiological  correlate  of  cognition  in  general.  For 
the  cognitive  determination  of  temporal  relations,  or 
relations  of  agreement  and  difference,  etc.,  is  to  be  re- 
garded as  a  case  of  the  further  determination  of  relations 
which  were  earlier  apprehended  in  a  comparatively 
indeterminate  form.  Now  the  actual  experiences,  and 
consequently  the  corresponding  neural  processes,  are 
in  fact  determinate  in  character;  so  also  are  the  relations 
amongst  them.  How,  then,  can  this  determinateness 
of  character  and  of  relation  be  combined  with  varying 
degrees  of  indeterminateness  that  should  be  correlated 

8—2 


ii6 


CHAPTER  VIII 


with  the  varying  degrees  of  indeterminateness  of  cog- 
nition ? 

§  7.  It  appears  to  me  that  the  reason  why  physio- 
logists and  psychologists  never  properly  face  the  problem 
of  the  neural  correlate  of  cognition,  is  because  they 
virtually  identify  ideas  with  images.  This  confusion  is 
especially  apparent  in  discussions  about  so-called 
'general  ideas,'  'abstract  ideas,'  'conception,'  or  'con- 
ceptual thinking.'  The  older  nominalists  denied  the 
possibility  of  such  forms  of  thought  or  ideation,  and 
maintained  that  the  only  mental  content  which  can  actu- 
ally be  asserted  in  abstract  thinking  is  the  word  heard, 
or  uttered,  or  represented  in  imagery.  For  modern 
psychology  the  problem  of  the  relation  between  language 
and  thought  is  still  a  burning  question.  So  far  as  lan- 
guage is  concerned,  the  simplest  case  to  take  for  illus- 
tration would  be  that  in  which  we  characterise  a  sense- 
experience,  say  as  being  red,  and,  in  this  act  of  charac- 
terisation, utter  or  image  the  word  red.  This  process  is 
partly  explained  by  the  formula  of  association,  and  the 
associative  process  may  be  safely  assumed  to  have  a 
neural  correlate ;  so  that  the  mental  association  between 
the  sensation  red  and  the  word  red  may  be  correlated 
with  some  connective  process,  taking  place  between  the 
central  or  cerebral  processes  which  underlie  the  visual 
sensation  and  the  word-imagery  respectively.  But  even 
in  this  simplest  case,  it  must  be  pointed  out,  that  to 
define  the  mental  process  as  merely  an  association  be- 
tween the  word  and  the  sensation,  is  wholly  inadequate. 
On  the  occurrence  of  the  sensation,  it  is  not  only  the 
image  of  the  word  which  is  aroused  in  the  mind  of  the 
thinker;  but  he  mentally  connects  the  word  with  the 


»■/ 


4t 


K* 


41   • 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     117 

sensation  in  a  form  which  could  be  expressed  in  some 
such  proposition  as:  *The  word  red  stands  for  the 
quality  characterising  this  sensation.'  In  fact,  passing 
from  this  simplest  case  to  the  higher  forms  of  thought 
which  may  be  accompanied  by  language,  the  verbal 
expression  of  a  proposition  may  be  taken  to  represent 
the  universal  form  of  an  act  of  thinking.  Association 
and  its  probable  neural  correlate  are  involved  in  so  far 
as  the  words,  comprising  the  entire  sentence,  can  be 
said  to  be  associated  with  one  another  and  with  the 
objects  to  which  they  refer,  so  as  to  constitute  a  whole. 
But,  if  we  examine  the  mental  process,  we  find  that  the 
sentence  is  not  merely  a  whole  for  the  thinker,  but  a 
significant  whole.  Mere  association  might  give  an 
adequate  account  of  a  combination  of  words  which 
was  mere  nonsense;  it  cannot  account  for  the  added 
psychical  fact  that  the  sentence  is  understood  as  having 
meaning.  Now  an  understanding  of  the  meaning  of 
language,  and  in  particular  of  the  sentence  as  denoting  a 
proposition,  is  what  is  meant  by  thought  or  ideation.  If 
the  physio-psychical  process  of  association  is  adequate  to 
explain  how  the  words  of  the  sentence  come  successively 
before  the  mind,  what  can  be  the  physiological  process 
correlated  with  the  act  of  understanding  its  significance? 
§  8.  The  distinction  between  physio-psychical  pro- 
cesses and  those  which  I  attribute  to  subjective  initiation 
is  best  illustrated  by  those  reflex  processes  which  entail 
various  forms  of  consciousness,  for  here  the  contrast  is 
sharp  and  unmistakable.  Let  us  take  for  example  the 
condition  which  we  speak  of  as  a  tendency  to  sneeze ; 
this  is  a  physiological  condition  which  is  certainly  ac- 
companied by  several  definable  phases  of  consciousness: 


r^  -> 


ii8 


CHAPTER  VIII 


( 1 )  There  is  in  the  first  place  a  characteristic  sen- 
sation which  is  distinguishable  for  instance  from  that 
accompanying  the  physiological  tendency  to  cough. 

(2)  Like  other  sensations,  at  the  time  of  its  experi- 
ence its  character  may  be  cognitively  defined  both  as 
regards  quality  and  locality  by  the  experient;  and  it 
will  be  so  defined  if  any  interest  or  purpose  prompts 
him  to  direct  his  attention  to  it. 

(3)  So  long  as  the  physiological  tendency  to  sneeze 
remains  a  tendencj^,  there  is  an  element  of  feeling  which 
could  be  called  discomfort. 

(4)  There  is  normally  anticipatory  imagery  or  pre- 
cognition of  what  will  almost  immediately  occur  sen- 
sationally when  the  sneeze  actually  takes  place.  This 
foreknowledge  of  the  prospective  sensation  involves  au- 
ditory and  other  forms  of  imagery. 

We  may  now  pass  from  what  is  a  direct  analysis  of 
the  modes  of  consciousness  accompanying  the  physio- 
logical tendency,  to  a  consideration  of  the  causal  pro- 
cesses which  may  follow.  In  the  absence  of  any  in- 
hibitory act  on  the  part  of  the  experient,  the  sneeze  will 
take  place ;  and  in  this  case  the  causality  involved  is 
purely  physical  or  physiological.  But  the  sneeze  has  a 
sensational d.s  well  as  a  physiological  aspect;  and  to  this 
aspect  the  term  conative  has  been  mistakenly  applied ; 
the  term  conation  being  taken  as  equivalent  to  felt 
tendency.  But  this  definition  appears  to  me  to  involve 
confusion;  for  the  phrase  *felt  tendency'  has  been  used 
to  describe  the  form  of  conscious  experience  which  ac- 
companies a  process  which  will  normally  terminate  in 
an  explosion;  whereas  it  is  generally  agreed  that  cona- 
tion is  a  form  of  consciousness  of  which  the  causality  is 


>> 


4V 


T' 


V 


■•■>■ 
A" 

4 


4.. 


m. 


K 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     119 

(  psychical  and  not  merely  physiological.  I  n  the  case  of 
the  sneeze,  the  word  'tendency'  alone  would  express  a 
merely  physiological  fact;  and  when  conjoined  with  the 
word  felt,  it  can  signify  only  a  precognition  of  the 
subsequent  mode  into  which  the  sensational  process 
accompanying  the  physiological  fact  will  develop.  In 
the  physiological  case,  discomfort  and  foreknowledge 
play  no  part  as  causal  factors  in  determining  the  further 
process ;  psychical  causality  would  only  enter  therefore 
in  an  effort  to  inhibit,  repress,  retard,  or  moderate  the 

(  sneeze.  The  exercise  of  this  effort  is  conditioned  by 
precisely  the  same  factors  as  operate  in  any  case  of 
voluntary  effort;  namely  by  desire  for  a  more  or  less 
specifically  cognised  end,  and  a  knowledge  of  the  im- 
mediate means.  In  the  case  before  us,  the  conative 
process  proper  operates  in  resisting  or  opposing  a  purely 
physical  or  physiological  tendency :  it  might  be  com- 
pared with  the  effort  of  a  man  to  open  an  umbrella 
in  a  strong  wind.  My  purpose  in  the  foregoing  analysis 
is  to  distinguish  this  type  of  case,  where  conation  ope- 
rates in  opposition  to  a  merely  physiological  tendency, 
from  that  in  which  one  conative  tendency  conflicts  with 
another.  These  two  cases  are  apt  to  be  fatally  con- 
fused ;  for  they  agree  superficially  not  only  in  the  point 
that  one  tendency  operates  in  opposition  to  another, 
but  also  in  the  further  point  that  the  struggle  between 
the  operations  of  the  two  tendencies  is  normally  ac- 
companied by  an  additional  psychical  factor  of  feeling — 
either  mere  discomfort,  or  a  highly  intense  feeling 
amounting  to  pain :  e.g.  any  moderate  degree  of  discom- 
fort which  may  precede  the  actual  sneeze  is  considerably 
heightened  in  the  process  of  endeavouring  to  suppress  it. 


120 


CHAPTER  VIII 


y^ 


A  case  which  resembles  more  closely  still  the  voluntary 
attempt  to  inhibit  a  reflex  process  is  the  stimulation  of  an 
emotion ;  but  here  the  parts  played  by  the  physical  and 
psychical  factors  are  reversed.  Thus  when  cognition 
of  the  circumstances  has  aroused  conation,  and  this  has 
naturally  developed  into  specifically  directed  purpose, 
there  occurs  what  is  termed  an  emotional  experience  if 
this  directed  purpose  is  accompanied  by  irrelevant  or 
perturbing  organic  processes,  which  may  be  assumed 
to  be  partially  reflex.  Thus,  in  the  case  of  emotion, 
the  physiological  causality  is  apt  to  interfere  with  the 
psychical  causality  manifested  in  the  form  of  purpose; 
whereas  in  the  case  of  the  sneeze  it  is  the  purpose 
which  interferes  with  the  reflex  process. 

§  9.  We  may  now  turn  our  attention  to  the  highly  com- 
plex processes  of  conative  conflict,  and  for  the  purpose 
of  this  analysis  I  shall  introduce  a  simple  mode  of  sym- 
bolism. It  may  be  taken  to  be  the  general  case  that 
when  there  are  two  alternatives  either  of  which  can  be 
actualised,  there  will  be  in  each  alternative  some  aspect 
or  circumstance  which  is  attractive,  and  some  other  as- 
pect or  circumstance  which  is  repellent.  If  the  aspects 
felt  as  attractive  be  represented  by  the  symbols  a  and  b, 
and  those  felt  as  repellent  by  the  symbols  a'  and  b\  then 
ab'  will  stand  for  one  alternative,  and  a'b  for  the  other; 
and  the  conflict  will  be  symbolised : 

ab^  versus  a^b. 
It  is  of  psychological  importance  to  regard  the  two  sym- 
bols a  and  a'  as  positively  opposed  modes  of  the  same  ge- 
neric or  determinable  aspect;  because  the  mere  negative 
non-a  could  not  be  represented  in  thought  or  imagery 
with  a  felt  repulsion  or  attraction ;  what  arouses  conation 


..  1 


tf 


\r» 


^    f 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     121 

must  have  some  positive  content.  Only  in  the  limiting 
case,  therefore,  where  the  felt  conation  was  indifferent 
and  could  be  measured  as  zero,  could  we  apply  the 
symbol  a^  or  b'  to  denote  merely  non-a  or  non-b.  In 
order  to  measure  the  felt  conation  we  will  use  the  Greek 
letters  corresponding  to  the  Latin,  with  the  sign  ( + )  to 
stand  for  the  aspect  felt  as  attractive,  and  the  sign  ( —  ) 
for  the  aspect  felt  as  repellent.  Then,  if  the  symbol  c 
stands  for  the  felt  intensity  of  the  conation. 


c  oi  a=-{-a'j 


c  of  a!—  "d  ; 


c  oi  b^-\r^\  c  of  b'=-^. 

The  resultant  conation  of  each  of  the  alternatives  is  de- 
termined jointly  by  its  attractive  and  repellent  aspects : 
it  may  be  shortly  symbolised  as  re.    We  thus  reach  the 


results: 
Hence: 


/- 


re  of  a  with  b'—a-^\  re  oi  b  with  d—^-  a. 

ace.  as  a  -  jS'  >  or  <  i3  -  a, 
i.e.  ace.  as  a-k-d  >  or  <  ^  +  /3', 

so  is  re  of  ab'  >  or  <re  of  a'b. 

* 

We  shall  of  course  assume  that  the  stronger  resultant 
conation  is  that  which  wins  in  the  conflict.  Now  there 
are  occasions  when  the  subject  is  indifferent  about  the 
issue  of  two  conflicting  conations.  For  example,  if  I 
am  in  doubt  which  to  choose  of  two  possible  entertain- 
ments, say  a  concert  or  a  game  of  bridge,  then  I  may 
be  deliberating  merely  in  the  sense  that  I  allow  the 
attractive  and  repellent  aspects  of  the  two  alternatives, 
as  represented  in  imagination,  to  work  themselves  out, 
without  reference  to  consequences  or  other  considera- 
tions. Deliberative  process  of  this  kind,  in  which  the  in- 
ner working  of  conative  tendencies  is  passively  watched, 
may  be  called  non-moral  deliberation.    When,  on  the 


122 


CHAPTER  VIII 


other  hand,  the  subject  is  not  indifferent  as  to  which  of 
the  conations  will  prevail,  he  would  seem  to  have  the 
power  to  decide  the  issue.  The  desire,  or  conative  ten- 
dency which  may  be  said  to  urge  him  to  exert  this  power, 
and  thus  to  modify  the  strength  of  his  primary  desires 
or  conations,  may  be  called  a  secondary  (or  perhaps 
moral)  desire  or  conation;  and  we  will  now  examine 
the  practical  means  by  which  this  secondary  process  is 
exercised.  On  the  supposition  that  it  is  the  first-men- 
tioned alternative  that  is  to  be  enforced,  it  will  be  seen 
that  what  is  required  is  to  strengthen  the  felt  attraction 
to  a  and  the  felt  repulsion  to  a\  and  at  the  same  time 
to  weaken  the  felt  attraction  to  b  and  the  felt  repulsion 
to  b\  Thus  it  is  not  only  the  attraction  to  a,  but  also 
the  repulsion  to  its  alternative  a^  which  must  be  felt 
more  strongly,  if  the  will  is  to  accept  a  and  reject  a^\ 
while  if  b  is  to  be  rejected  in  spite  of  its  attractiveness, 
and  b^  to  be  accepted  notwithstanding  its  objection- 
ableness,  then  the  felt  force  of  these  factors  must  be 
weakened.  I  attach  the  highest  importance  to  this 
double  reference  to  the  repulsive  as  well  as  to  the 
attractive  tendencies;  and  on  this  point  it  appears  to 
me  that  James — whose  symbolic  representation  of  the 
process  agrees  to  some  extent  with  mine — wrongly 
represents  the  matter.  I  agree  with  him  that  an  act 
of  attention,  which  renders  more  vivid  the  imagery  or 
more  determinate  the  idea  connected  with  the  various 
aspects  of  the  two  alternatives,  is  the  essential  factor 
in  the  process.  But  in  my  opinion,  James  misrepresents 
the  case  when  he  assumes  that  the  single  effect  of  render- 
ing to  oneself  an  aspect  more  vividly  is  to  strengthen 
the  inclination  to  actualise  that  aspect.    This  can  surely 


A,X 


I 


■<• 


w' 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     123 

only  be  the  case  when  the  aspect  strengthened  by  at- 
tention is  itself  attractive ;  for  if  it  is  repulsive,  then  the 
effect  of  attention  is  to  strengthen  the  felt  repulsion. 
In  point  of  fact,  in  many  cases  of  moral  conflict,  it  is 
the  more  determinate  thought  of  what  will  in  some 
sense  pain  us  in  the  alternative  which  our  moral  atti- 
tude directs  us  to  reject,  that  helps  us  to  decide;  rather 
than  the  more  determinate  thought  of  what  will  please  us 
in  the  alternative  which  the  moral  attitude  prompts  us  to 
accept.  In  the  general  case  we  shall  include  both  factors. 
Now  this  power  of  modifying  primary  desires  and 
aversions  by  direct  attention  is  one  of  the  most  con- 
spicuous forms  in  which  the  ordinary  man  claims 
freedom  of  the  will.  Since  the  mere  uninfluenced  force 
of  his  desires  and  aversions  does  not  inevitably  deter- 
mine the  issue  and,  by  exerting  the  power  he  possesses 
over  the  direction  of  his  attention,  he  is  able  to  influence 
the  ultimate  decision,  he  holds  that  therefore  there  is  free 
will.  This  very  important  sense  in  which  freedom  of 
the  will  has  to  be  maintained  does  not  infringe  the 
principle  of  causal  determination  which  we  attribute  to 
volitional  processes :  it  is  not  a  mere  accident  or  matter 
of  chance  whether  the  secondary  desire  does  arise  with 
a  strength  sufficient  to  change  the  issue ;  the  agent  s 
secondary  desires  will  have  had  a  set  of  antecedent 
causal  conditions  similar  to  those  which  we  ascribe  to 
the  primary  desires.  Tracing  this  causal  chain  back- 
wards, we  have  of  course  to  presume  potential  conative 
tendencies  present  at  the  early  periods  of  the  develop- 
ing experience;  and  science  is  here,  as  in  other 
branches  of  psychology,  supremely  ignorant,  the  actual 
causal  conditions  constitutingaproblem  for  investigation. 


124 


CHAPTER  VIII 


§  lo.  In  this  analysis  I  have  referred  exclusively  to 
the  conative  aspect  of  deliberative  volitional  processes ; 
but  logically  speaking,  it  would  have  been  perhaps  more 
correct  to  define  the  cognitive  processes  before  analys- 
ing the  conative ;  for  the  judgment  or  knowledge 
which  is  presupposed  when  we  speak  of  the  alternatives 
within  the  range  of  possible  actualisation  by  the  agent, 
is  totally  independent  of  the  conative  aspects.  It  is 
only  after  judgment  with  respect  to  the  possible  ends 
has  been  so  to  speak  impartially  exercised,  that  the 
conative  forces  moving  in  one  or  the  other  direction, 
begin  to  operate.  While,  therefore,  the  cognitive 
processes  without  the  conative  would  be  inadequate 
as  determinants  of  the  will,  just  as  would  the  conative 
without  the  cognitive,  it  is  nevertheless  true  to  say  that 
the  forms  assumed,  and  the  prior  conditions  which 
account  for  the  forms  assumed  by  the  judgment,  are 
totally  and  absolutely  independent  of  the  conative 
processes.  Psychologists  who  have  given  a  clear 
enough  analysis  of  cognitive  deliberation  have  made, 
what  appears  to  me  to  be,  the  fatal  mistake  of  attempt- 
ing to  reduce  conative  deliberation  to  the  same  type  of 
formula.  The  cognitive  aspect  of  a  deliberative  process 
is  concerned  merely  with  the  known  or  accepted  causal 
conditions  for  actualising  any  supposed  end,  and  this 
process  of  judgment  has  the  same  conative  imparti- 
ality as  any  scientific  problem — theoretic  or  practical. 
That  this  cognitive  or  intellective  process  is  to  be 
assumed,  seems  to  me  to  raise  no  controversy.  I  have 
therefore  laid  emphasis  upon  the  conative  aspect  of 
deliberation  since  I  hold  that  it  is  to  this  process  that 
causality  essentially  applies.   In  this  view  I  am  opposed 


r' 


> 


4it 


Jl 


H 


V 


V 


4» 


APPLICATION  OF  CAUSAL  NOTIONS  TO  MIND     125 

to  those  psychologists  who  maintain  that  the  will  is  free 
in  the  sense  that  it  can  act  on  mere  judgment  without 
any  conative  urging  ;  and  it  is  upon  this  issue  that  the 
burning  controversial  problem  essentially  depends. 

§  II.  The  above  general  reference  to  judgment  as 
essential  in  the  higher  volitional  processes  must  be  sup- 
plemented by  a  consideration  of  the  different  kinds  of 
predicates  and  their  correlated  subjects  which  together 
constitute  the  various  types  of  propositional  content  So 
far  the  judgments  entering  into  the  process  of  delibera- 
tion to  which  we  have  alluded  have  been  those  directed 
to  physical  or  at  any  rate  external  conditions,  predicating 
of  these,  characters  quite  independent  of  mental  refer- 
ence. But  the  judgment  which  distinguishes  the  higher 
human  volitions  attributes  value  to  possible  existents, 
and  in  certain  relevant  cases  comparative  values  to  dif- 
ferent alternatives ;  such  judgments  predicating  of 
their  objects  characters  which  are  intrinsic  to  them,  in 
the  sense  that  they  are  entirely  independent  of  the  likes 
and  dislikes  of  the  person  judging.  Without  entering 
into  controvertible  issues,  it  will  be  universally  admitted 
that  when  objects  are  characterised  by  such  adjectives 
as  good  or  beautiful,  they  can  properly  be  said  to  be 
raised  into  a  realm  of  reality  removed  from  that  realm 
in  which  reference  is  made  merely  to  predicates  based 
upon  qualities  of  sensation,  or  upon  the  scientifically 
developed  properties  of  continuants.  At  any  rate  these 
adjectives  'good'  and  'beautiful'  are  imposed  upon  their 
objects  in  an  act  which  is  quite  other  than  the  analytico- 
descriptive  characterisations  made  by  what  we  may  call 
science ;  and  apart  altogether  from  any  influence  upon 
volition,  this  species  of  judgment  has  unique  features, 


126 


CHAPTER  VIII 


<' 


which  distinguish  it  from  the  type  of  judgment  with 
which  the  simpler  logic  is  mainly  concerned.  With 
regard  to  the  influence  of  such  judgments  upon  cona- 
tion, it  may  be  that  an  attitude  is  necessarily  evoked 
which  tends  to  stimulate  the  thinker  to  produce  so 
far  as  possible  the  kind  of  object  to  which  value  is 
attached  in  his  judgment.  If  so,  a  judgment  of  value 
of  this  kind  may  be  said  to  be  by  itself  the  sufficient 
cause  of  a  direct  act  of  will.  Where,  on  the  other  hand, 
the  judgment  is  not  accompanied  by  a  felt  urgency 
sufficiently  strong  to  overcome  conflicting  tendencies, 
it  may  still  be  a  pure  judgment  of  value.  What,  in  my 
opinion,  constitutes  the  importance  of  judgments  of  this 
kind,  is  that  where  any  causal  relation  between  the 
judgment  and  the  conation  subsists,  it  is  the  character 
predicated  which  causes  the  conation,  and  not  con- 
versely, the  conation  or  felt  tendency  to  actualise  the 
object,  which  causes  the  judgment  of  value. 


J. 


<'- 


~) 


<> 


4» 


CHAPTER  IX 

TRANSEUNT  AND  IMMANENT  CAUSALITY 

§  I.    To  understand  the  distinction  between  transeunt 
and  immanent  causality  it  is  necessary  to  have  grasped 
the  conception  of  the  continuant ;  and  to  illustrate  how 
the  continuant  functions  in  this  connection,  it  will  be 
convenient  once  again  to  analyse  what  is  meant  in 
physical  science  by  movement.    We  may  speak  for  in- 
stance of  points   A    and   B   being   occupied  at  one 
instant  of  time,  and  the  points  C  and  D  unoccupied; 
while  at  a  subsequent  instant,  points  C  and  D  are 
occupied.     In  the  temporal  interval  from  one  instant 
to  another  something  physical  has  happened  to  which 
the  name  movement  is  given.    But  such  movement 
cannot  be  unequivocally  described  unless  we  are  able 
to  distinguish  between  two  such  cases  as  first  a  move- 
ment from  ^  to  C  and  from  B  to  Z?,  and  second  a 
movement  from  A  to  D  and  from  B  to  C.     Unless  we 
know  which  of  these  alternatives  is  the  correct  descrip- 
tion, our  conception  of  what  has  happened  in  the  time- 
interval  is  undefined,  and  no  subsequent  events  can  be 
inferred  without  presuming  one  or  other  of  these  alter- 
natives ;  so  that,  in  constituting  the  event  called  move- 
!  ment,  we  must  assume  something  that  moves,  to  which 
the  name  particle  may  here  be  given.    The  one  alter- 
native then  is  that  a  particle  has  moved  from  A  to 
C  while  another  particle  has  moved  from  B  to  D\  and 
the  opposite  alternative  that  a  particle  has  moved  from 


V. 


128 


CHAPTER  IX 


A  to  D  while  another  particle  has  moved  from  B  to  C. 
And  when  we  speak  of  two  particles  we  conceive  of  each 
as  continuing  to  exist  and  as  retaining  its  identity  with 
itself  and  its  diversity  from  the  other;  so  that  terms 
such  as  this,  or  that,  or  it,  involving  reference  to  the 
same  particle,  are  required  to  describe  what  we  hold  to 
be  the  character  of  the  event. 

§  2.  From  this  elementary  illustration  of  a  continuant, 
we  may  pass  directly  to  an  illustration  of  immanent 
causality.  Thus,  when  the  movement  of  a  particle  from 
A  to  B  during  an  interval  of  time  is  followed  by  a  move- 
ment of  the  same  particle  from  B  to  K,  the  law  or  for- 
mula in  accordance  with  which  the  nature  of  the  former 
movement  determines  that  of  the  latter  exhibits  imma- 
nent causality;  i.e.  the  causality  in  which  the  cause 
■occurrence  and  the  effect  occurrence  are  attributed  to 
the  same  continuant.  The  law  in  this  case  is  known  as 
the  first  law  of  motion,  and  it  can  be  briefly  expressed 
thus:  the  speed  and  direction  of  the  movement  of  a 
particle  is  maintained  unchanged  from  one  period  of 
time  to  another.  The  empirical  establishment  of  this 
formula  presupposes  that  no  other  form  of  causality  in- 
tervenes. But  when,  in  the  physical  domain,  one  par- 
ticle is  regarded  as  causal  agent  and  the  other  as  patient, 
in  the  sense  that  the  movement  of  the  latter  is  conceived 
as  the  effect  occurrence,  while  the  position  of  the  former 
relatively  to  it  constitutes  the  cause  occurrence,  a  dif- 
ferent notion  of  causality  is  introduced,  and  this  we 
shall  call  transeunt.  Here  then  the  cause  occurrence  and 
the  effect  occurrence  are  referred  to  different  continuants, 
whereas  in  immanent  causality  cause  occurrence  and 
effect  occurrence  are  attributed  to  the  same  continuant. 


I 


/I 


4k 


«  tb. 


^r' 


t  ^ 


^^ 


I 

4> 


TRANSEUNT  AND  IMMANENT  CAUSALITY        129 

This  illustration  serves  further  to  indicate  what  may 
be  assumed  to  be  universally  applicable,  that  any 
concretely  described  causal  process  must  be  analysed 
into  a  conjunction  of  transeunt  and  immanent  causality; 
and  neither  types  of  causality  are  to  be  found  actually 
separate. 

§  3.  We  now  pass  to  a  somewhat  complicated  physical 
illustration;  namely  the  case  of  a  gas  of  which  the 
pressure,  volume  and  temperature  are  conceived  as  its 
alterable  states.  The  gaseous  body  to  which  we  here 
refer  is  not  a  simple  or  ultimate  physical  continuant 
such  as  a  particle,  but  consists  of  an  indefinite  number 
of  ultimate  constituents  to  each  of  which  the  name 
molecule  is  familiarly  attached.  The  spatial  relations 
amongst  these  molecular  constituents  are  alterable,  so 
that  the  gaseous  body  as  a  whole  may  be  said  to  have 
an  inside,  and  the  terms  pressure,  volume  and  tempera- 
ture are  therefore  permissible  as  defining  its  alterable 
states.  Before  we  proceed  to  consider  some  actual 
process  of  experimentation  upon  such  a  gaseous  body, 
it  may  be  pointed  out  that  experiment  itself  implies 
transeunt  causality ;  for  the  experimenter  employs  physi- 
cal agents  whose  movements  he  himself  controls,  and 
these  produce  in  the  material  operated  upon,  effects 
which  would  not  have  been  produced  apart  from  this 
external  manipulation.  If,  in  the  simplest  case  of  the 
laboratory,  the  bodies  there  occupying  space — being 
conceived  in  their  combination  as  a  whole — were  left 
to  themselves,  then  the  changes  which  would  take  place 
would  come  under  the  head  of  immanent  causality.  But 
when  the  experimenter,  by  manipulating  other  bodies 
which  he  can  control,  produces  effects  which  modify  the 

J  L  HI  9 


130 


CHAPTER  IX 


course  of  immanent  causality,  these  must  be  described 
as  transeunt. 

Let  us  first  consider  the  case  where  the  pressure  upon 
the  gas  remains  constant,  and  the  experimenter,  by 
means  of  controlling  instruments,  alters  the  temperature 
and  awaits  the  effect,  exhibited  as  a  change  of  volume, 
in  the  gaseous  body.  Here  the  mathematician  briefly 
expresses  the  formula  of  causality  in  the  equation/z/  =  kty 
where  /,  v,  and  /  represent  respectively  the  pressure, 
volume  and  temperature  of  the  gas,  and  ^  is  a  constant 
coefficient  which  measures  a  certain  unchanged  pro- 
perty specific  to  the  gas  under  experimentation'.  This 
familiar  mathematical  formula  is  inadequate,  however,  to 
express  the  joint  transeunt  and  immanent  causality  which 
we  propose  to  investigate.  I  shall  therefore  symbolise 
two  pressures,  temperatures  and  volumes,  one  applying 
to  the  gaseous  body  itself — which  I  shall  call  internal 
— and  the  other,  which  I  shall  call  external,  to  the  sur- 
rounding body  or  envelope,  the  suffixes  i  and  e  being 
employed  thus  A,  A>  ^«>  ^^»  ^i^  ^e,  to  represent  one  and 
the  other  continuant.  The  case  before  us  is  that  in 
which  gas  is  contained  in  a  cylinder,  the  volume  of  which 
is  free  to  change  as  the  weight  placed  on  the  top  changes. 
The  temperature  of  the  gas  /,•  is  then  determined  by  the 
temperature  t,  of  the  cylinder,  which  may  therefore  be 
regarded  as  agent  in  the  process  described  as  the  conduc- 
tion of  heat  which  causes  /,  to  equal  t,.  Here,  then,  t^ 
being  referred  to  the  surrounding  body  as  cause,  and 
li  to  the  gaseous  body  as  effect,  we  have  transeunt 
causality.  Similarly  the  weight  of  the  piston,  of  which 
the  experimenter  has  direct  control,  represented  by  the 
pressure  p,,  determines  the  pressure  pi  in  accordance 

^  See  Part  II,  Chapter  V,  §  9. 


\ 


4v 


41 


r     ^ 


''  w 


4.N 


^)> 


■u 


TRANSEUNT  AND  IMMANENT  CAUSALITY        131 

with  the  process  described  as  the  transference  of  pres- 
sure which  causes/,  to  equal /^,  Here,  then,  there  are 
two  separately  conjoined  transeunt  causal  factors/^  and 
tgy  each  of  which  entails  as  a  separate  effect  pi  and  /^. 
Next  the  conjunction  of  the  two  factors/,  and  /,  charac- 
terising the  gas  itself,  causally  determines  the  effect  z/,-, 
in  accordance  with  the  nature  of  the  gas,  thus  exhibiting 
immanent  causality.  Lastly  z;,-,  which  is  the  volume  of 
the  gas,  determines  as  its  effect  the  volume  occupied  by 
the  cylinder  which  will  be  v^,  since  the  expansive  pro- 
perty of  gas  causes  v^  to  equal  Vi.  In  other  words,  this 
last  causal  process  is  again  transeunt,  but  it  is  from  the 
gas  as  agent  to  the  cylindrical  envelope  as  patient.  The 
whole  process,  then,  may  be  schematised  as  follows: 

Transeunt        Immanent        Transeunt 


Vi  -*- 


kt 
under  the  formula  (say)  7/*=—*  (immanent). 

Pi 

and  ti^t^\  pi=p^\  Vi=Vi   (transeunt). 

§  4.  A  new  problem  is  at  once  suggested  by  the  above 
immanent  formula,  which  connects  together  the  alterable 
volume,  temperature  and  pressure  of  the  gas,  showing 
them  to  be  related  independently  of  any  action  upon 
the  gas  from  without.  In  Part  II  under  the  heading  of 
functional  deduction,  the  notion  of  the  reversibility  of 
cause  and  effect  was  treated  in  its  mathematical  or 
deductive  aspect.  This  same  principle  is  illustrated  in 
the  case  before  us  by  taking  indifferently  /  and  /  as 
independent  of  one  another,  and  v  as  dependent  jointly 
upon  them ;  or  /  and  v  as  independent  of  one  another, 
and/  as  dependent  jointly  upon  them ;  and  we  have  now 
to  reconsider  the  principle  under  its  inductive  and  causal 
aspect.    In  the  above  account  of  the  experiment,  the 

9—2 


v» 


132 


CHAPTER  IX 


volume  of  the  gas  is  represented  as  effect,  and  its  tem- 
perature and  pressure  as  the  two  cause-factors;  but  the 
problem  arises,  since  the  factors/  and  v  are  manifested 
simultaneously,  how  to  determine  which  of  them  is  to 
be  called  cause  and  which  effect;  and  indeed  philosophical 
criticism  of  the  conception  of  causality  frequently  sug- 
gests the  view  that  where  the  cause  occurrence  and  the 
effect  occurrence  are  represented  as  simultaneous,  there 
is  no  principle  for  deciding  which  of  the  two  occurrences 
constitutes  cause  and  which  effect.  Now  the  general 
principle  whereby  I  distinguish  the  cause  from  the  effect 
where  manifestations  are  temporally  coincident,  is 
developed  from  the  distinction  and  connection  between 
immanent  and  transeunt  causality.  To  explain  this 
point,  let  us  turn  from  the  above  described  experiment 
of  a  cylinder  which  is  allowed  to  increase  in  volume  by 
a  movable  piston,  to  one  in  which  the  experimenter 
encloses  the  gas  in  an  inextensible  envelope.  The  two 
experiments  will  agree  in  respect  of  the  temperature 
process,  i.e.  in  either  case  the  surroundings  are  at  a 
certain  temperature  t^  which,  through  conduction,  will 
produce  a  temperature  /,•  of  the  gas  equal  to  t,.  But 
whereas  in  the  previous  experiment,  the  change  in 
volume  Vi,  produced  by  the  increased  temperature, 
causally  determined  v^,  in  the  new  experiment,  where 
the  volume  of  the  receptacle  is  controlled  by  the  experi- 
menter, V,  causally  determines  z/,-.  The  whole  process 
in  this  case  may  be  schematised  just  as  in  the  other,  by 
merely  interchanging/  and  v,  as  follows: 

Transeunt        Immanent        Transeunt 

Pi         -*■       A 


\ 


) 


4) 


ij 


•        V 

<  ■ 


<> 


». 


TRANSEUNT  AND  IMMANENT  CAUSALITY        133 

Now  in  comparing  the  two  schematisations  from  the 
point  of  view  of  the  causal  process  to  be  explained,  it 
may  be  asked  why,  while  /,  constitutes  a  cause-factor 
in  both  cases,  yet  in  the  first  case/,  is  said  to  function 
as  the  other  cause-factor  and  v^  as  the  effect,  in  the 
second  case  v^  is  said  to  function  as  the  other  cause- 
factor  and  /,•  as  the  effect.  Why  is  it  that  in  one  case 
pi  to  v^  stands  as  effect  to  cause,  and  in  the  other  case 
as  cause  to  effect.'^  In  the  first  experiment  v^wsls  not 
the  cause  of/,,  because  /^  was  the  cause  of/,-;  in  the 
second  experiment  /,•  was  not  the  cause  of  z^,,  because 
v^  was  the  cause  of  v^.  So  long  as  we  are  only  concerned 
with  the  alterable  states  of  the  one  continuant — i.e.  so 
long  as  we  are  concerned  only  with  immanent  causality 
— there  is  absolutely  nothing  to  determine  which  among 
the  co-variable  states  is  to  be  called  cause  and  which 
effect.  But  as  soon  as  we  refer  to  the  surrounding 
influences,  and  thereby  take  into  consideration  transeunt 
causality,  then  that  state  of  the  gas  which  is  the  imme- 
diate effect  of  the  state  of  the  surroundings  stands  as 
cause  relatively  to  the  other  inner  states  of  the  gas. 
Briefly  we  may  repeat  the  schemes  omitting  the  points 
in  which  they  agree:  in  the  first  experiment/^  causes 
/,,  and  therefore  it  is  /,•  that  causes  v^;  in  the  second 
experiment  v^  causes  z;,.,  and  therefore  it  is  v^  that  causes 
/,-.  This  is  generalised  in  the  following  principle :  when 
the  co-variable  states  of  a  body  are  causally  determined 
in  accordance  with  an  immanent  formula,  then  those 
variables  which  are  separately  effect-factors  in  the  tran- 
seunt process  must  constitute  the  cause-factors  jointly 
in  the  immanent  process. 

It  is  necessary  here  to  point  out  that  a  totally  different 


^>  r 


134 


CHAPTER  IX 


account  of  the  transeunt  and  immanent  processes  which 
the  gas  undergoes  is  required  when  (in  accordance  with 
the  kinetic  theory)  we  conceive  temperature  as  mole- 
cular kinetic  energy,  etc.  For,  here,  the  free  linear  move- 
ment of  each  molecule  illustrates  the  process  immanent 
to  that  molecule;  and,  at  the  instant  of  its  collision  with 
another  molecule,  is  illustrated  an  inter-molecular  tran- 
seunt process.  Moreover,  in  this  analysis,  the  events 
are  conceived  as  successive  and  not  simultaneous.  This 
more  ultimate  account,  however,  in  no  way  impairs  the 
validity  of  the  above. 

§  5.  It  may  be  helpful  to  pass  from  this  purely  physical 
illustration  to  a  case  of  psycho-physical  causality  which 
has  been  much  discussed  in  recent  times ;  viz.  as  to  the 
relation  of  an  emotion  to  its  so-called  expression.  Before 
the  James-Lange  theory  was  propounded,  emotion  was 
conceived  merely  as  a  mode  of  feeling  determined  es- 
sentially by  the  cognition  of  a  situation  as  such  or  such. 
This  analysis  disregarded  the  perturbing  concomitants 
of  such  experiences  as  those  of  fear  and  anger;  thus 
while  in  two  such  contrasted  experiences,  the  nature  of 
the  situation  was  held  to  be  an  object  of  differentiated 
cognition  on  the  part  of  the  experient— a  differentia- 
tion which  accounted  for  the  corresponding  difference 
in  the  purposive  acts  which  ensued — yet  in  the  analysis 
of  the  two  experiences,  no  place  was  given  to  perturb- 
ing organic  processes,  which  were  regarded  as  a  mere 
bye-product  of  the  emotional  state.  Now  James  held, 
in  my  opinion  correctly,  that  the  apprehension  of  a  situa- 
tion of  danger  which  leads  merely  to  adaptive  purposive 
action,  does  not  constitute  a  state  of  fear;  nor  would  a 
situation  in  which  an  experient  judged  himself  to  have 


1 


r 


<•   -- 


I 


•/ 


<#*   •. 


^> 


<\ 


<  » 


<\ 


) 


^,,4 

^     ^ 


%» 


TRANSEUNT  AND  IMMANENT  CAUSALITY        135 

been  injured  or  insulted  constitute  a  feeling  of  anger, 
if  it  simply  led  to  correspondingly  purposive  action. 
A  mere  cool  intellectual  judgment — which  is  not  alto- 
gether a  human  impossibility — could  not  properly  be 
called  an  emotional  state;  in  order  to  constitute  an 
emotion,  the  presence  of  perturbing  organic  processes 
which  produce  corresponding  organic  sensations  is  an 
essential — or  even  in  James'  view  apparently  the  essen- 
tial— condition  for  the  emotional  consciousness.  In  my 
opinion  the  only  error  in  James'  view  is  that,  while  it 
is  true  that  the  cognition  of  the  situation  without  ac- 
companying organic  disturbances  would  not  constitute 
an  emotional  state,  yet  neither  would  the  organic  pro- 
cesses constitute  an  emotion  apart  from  some  corre- 
sponding form  of  cognition  of  the  situation.  Emotion 
requires  the  presence  of  both  cognitive  and  sensational 
factors. 

Agreeing  so  far  with  James'  account  of  the  con- 
comitant factors  concerned  in  an  emotional  experience, 
we  proceed  to  consider  how  the  various  elements  are 
causally  related.  If  it  be  admitted  that  the  immediate 
initiative  of  an  emotional  experience  is  the  cognition  of  a 
situation  as  being  of  a  certain  kind,  then  this  cognition 
partially  illustrates  transeunt  causality,  inasmuch  as 
something  presented  to  perception  or  imagination  oc- 
casions the  content  and  form  of  the  cognition.  The  cog- 
nition to  which  a  separate  cause  has  thus  been  assigned, 
must  be  assumed  to  arouse  a  conative  process,  for  on  no 
other  hypothesis,  I  think,  can  emotion  be  explained ;  and 
this  conative  process,  being  a  manifestation  of  the  nature 
of  the  experient,  illustrates  immanent  causality.  At  this 
point  we   again  assume    a  transeunt  psycho-physical 


^*  » 


136 


CHAPTER  IX 


process,  but  in  the  reverse  order — viz.  from  the  cona- 
tive  tendency  as  cause  to  the  physiological  process 
as  effect.  Thus  the  conative  tendency  being  aroused 
suddenly  or  with  intensity,  the  immediate  consequent 
physiological  effects  are  vaguely,  widely,  and  intensively 
diffused,  in  such  wise  that  the  physiological  disturbances 
accompanying  vastly  different  kinds  of  cognised  situa- 
tions have  many  factors  in  common.  These  emotive 
effects  are  to  be  sharply  contrasted,  in  my  view,  with 
the  purposive  effects  arising  out  of  the  thought  element 
involved  in  the  conation;  although  this  purpose,  like 
the  emotive  disturbances,  constitutes  a  psychical  cause 
of  a  physical  effect.  In  this  analysis,  the  conation  as 
a  psychical  process  is  seen  to  be  the  effect  of  an  ex- 
\  ternal  cause,  and  in  its  turn,  the  cause  of  whatever 
^  specific  occurrence  may  thereafter  take  place.  We  shall 
not  say,  therefore,  after  the  old  fashion,  either  that  the 
emotion  of  fear  causes  the  physiological  disturbances,  or 
that  the  physiological  disturbances  cause  the  emotion  ; 
for  the  emotion  is  not  simple,  but  a  compound  of  cog- 
nitive, conative  and  sensational  factors. 

§  6.  Having  illustrated  how  the  notions  of  transeunt 
and  immanent  causality  are  employed  in  physics,  we 
will  now  consider  the  much  more  complicated  case  of 
psychology,  assuming  the  philosophical  position  known 
as  dualism,  which  regards  the  psychical  continuant  as 
I  something  with  a  nature  fundamentally  distinct  from  that 
'  of  a  physical  continuant.  Just  as  the  one  indubitable  illus- 
tration of  a  physical  continuant  is  the  particle,  so  I  shall 
assume  that  the  only  indisputable  psychical  continuant  is 
an  experient,  or  (what  for  the  present  I  wish  to  take  as 
synonymous)  a  person,  a  mind,  a  self,  or  an  ego.   Now  the 


^ 


Y- 


/A 


f. 


-4 


TRANSEUNT  AND  IMMANENT  CAUSALITY    137 

person  and  the  particle  agree  in  being  what  I  call  a  con- 
tinuant, namely  something  which  continues  to  exist  and 
to  stand  in  relation  to  what  changes  in  the  course  of  time. 
The  continuant,  in  either  case,  must  be  distinguished  from 
its  property,  since  the  property  may  either  remain  un- 
changed, or  may  change  within  a  given  period  of  time ; 
the  terms  unchanging  and  changing,  therefore,  apply  to 
the  properties  (as  well  as  to  states  and  relations),  and 
not  to  the  continuant  itself.  Of  the  particle,  physicists 
maintain  that  it  has  one  property,  namely  mass,  which 
is  unchanged ;  and  the  abstract  dynamic  theory,  which 
has  held  the  field  almost  uninterruptedly  =^since  Newton, 
is  that  mass,  besides  being  the  only  unchanged  property 
of  a  particle,  is  the  only  property  that  can  be  attributed 
to  a  particle  as  such.  Whether  ultimately  all  particles 
have  the  same  mass ;  or  whether  different  particles  have 
different  masses,  is  irrelevant  to  our  present  purpose ; 
and  I  shall  assume  provisionally  that  in  either  case  mass 
is  a  property  of  the  particle,  which  is  to  be  distinguished 
from  any  complex  quantity  definable  in  terms  of  motion 
and  in  particular  of  acceleration ''^. 

Contrasting  the  notion  of  a  particle  with  that  of  the 
psychical  continuant  or  person,  we  may  say  of  properties 
attributable  to  any  given  person  in  the  first  place,  that 
they  will  not  agree  determinately  with  the  properties  of 
any  other  person ;  and  that  in  the  second  place,  they 
are  subject  to  change  in  the  course  of  time.  Thus  the 
concept  of  a  psychical  continuant  differs  from  the  out- 
set in  two  important  respects  from  that  of  the  physical 

*  The  modifications  of  Newtonian  physics  at  the  present  day 
strengthen— rather  than  weaken— the  form  of  my  logical  analysis: 
the  instructed  reader  may  easily  make  the  requisite  corrections. 


138 


CHAPTER  IX 


continuant,  in  that  (i)  no  property  of  a  psychical  con- 
tinuant is  determinately  the  same  as  that  of  another; 
and  (2)  that  no  determinate  property  attributed  to  a 
psychical  continuant  remains  unchanged  during  an  in- 
definite period  of  time. 

But  a  far  more  important  and  far  reaching  distinction 
between  the  two,  is  that  a  psychical  continuant  may  be 
said,  metaphorically  speaking,  to  have  an  inside,  while 
the  physical  continuant  has  not.  In  other  words,  apart 
from  the  property  of  mass,  nothing  can  be  predicated  at 
any  time  of  a  particle  except  its  position  relatively  to 
other  particles,  and  the  change  in  its  relative  position 
as  time  lapses;  "^  while  we  may  speak  properly  of  the 
complex  analysable  states  of  the  psychical  continuant. 
Thus  the  term  change,  when  applied  to  a  particle, 
refers  solely  to  the  external  relation,  position"^;  but 
when  applied  to  the  psychical  continuant  it  refers  pre- 
dominantly to  alteration  of  state.  It  is  true  that  science 
speaks  of  the  state  of  a  material  body,  where  the  body 
is  conceived  as  containing  a  number  of  particles;  but 
when  the  term  'temperature,'  for  example,  is  used 
to  denote  such  an  alterable  state  of  a  body,  it  means 
nothing  more  than  the  mode  of  agitation  of  the  particles 
which  constitute  the  body.  Since  this  mode  of  agitation 
(i.e.  of  relative  movement  of  the  particles  within  what 
in  many  cases  continues  to  be  the  same  body)  is  subject 
to  change,  the  conception  of  an  alterable  state  of  a  ma- 
terial body  is  legitimately  applicable  to  its  temperature. 
Other  conceptions  have  been  introduced  into  physical 
science,  resembling  temperature  in  these  respects;  i.e. 

*  This  again  requires  modification  in  the  light  of  present-day 
physical  science. 


/*.  .-' 


>» 


n 


TRANSEUNT  AND  IMMANENT  CAUSALITY        139 

they  represent  alterable  states  of  a  body,  logically 
distinguishable  both  from  the  aggregate  of  particles 
which  constitute  the  body  itself,  and  from  the  properties 
which  can  be  attributed  to  the  body  as  distinct  from  the 
mass  attributed  to  each  of  the  particles  of  which  it  is 

composed. 

The  property  of  a  composite  body— i.e.  a  body  com- 
posed of  an  aggregate  of  particles— is  to  be  defined, 
not  as 'anything  actually  manifested  in  time  and  space, 
but  as  a  rule  in  accordance  with  which  the  changes 
actually  manifested  in  time  and  space  can  be  formulated. 
Material  bodies  may  be  grouped  according  as  they  con- 
tinue to  be  composed  of  the  same  particles— such  bodies 
being  called  inorganic— or  according  as  they  are  com- 
posed of  minor  parts  differing  from  time  to  time,  these 
being  assimilated  from  external  bodies,  and  uniting  in 
modes  regulated  by  a  property  of  the  body,  i.e.  a  rule 
of  behaviour  formulating  the  processes  of  these  minor 
parts.    Such  bodies  are  called  organic. 

§  7.  The  further  consideration  of  the  character  of  psy- 
chical causality  leads  to  the  introduction  of  at  least  two 
fundamental  factors  uniquely  characteristic  of  mind, 
although  a  tendency  has  prevailed  throughout  the 
history  of  philosophy,  to  import  into  interphysical  re- 
lations these  conceptions  which  are  true  with  certainty 
and  immediateness  only  of  psychical  processes.  In  the 
\  first  place  to  mind  or  consciousness  or  experience,  I 
attribute  efficient  causality— a  notion  which  can  have 
but  dubious  application  in  the  physical  sphere,  while 
there  can  be  no  doubt  that  something  equivalent  to 
power  is  a  mental  factor  which  can  and  does  influence 
the  course  of  physical  events.    And  secondly,  there  are 


IT' 


140 


CHAPTER  IX 


r 


(the  processes  of  cognition,  judging  or  thinking,  which 
are  of  still  more  importance  both  as  causally  determi- 
native, and  because,  regarded  as  effects  manifested  as 
occurrences  in  time,  they  come  under  a  special  kind  of 
ontological  determinism.  Volition,  which  is  a  psychical 
act,  determined  jointly  by  conative  and  cognitive  pro- 
cesses, exhibits  both  these  peculiar  characteristics  of 
mind,  and  it  is  directly  determinative  of  physical  occur- 
rences. From  the  time  of  Aristotle,  the  second  unique 
aspect  of  mental  process  mentioned  above  has  been 
described  as  final  cause  ;  and  here,  as  elsewhere,  I  hold 
that  a  strictly  factual  account,  and  not  a  merely  meta- 
physical explanation,  is  to  be  given  of  this  notion.  In 
final  causality,  the  idea  of  E,  an  effect  or  end,  is  an 
j  essential  causal  factor  in  the  actualisation  of  E\  but  at 
the  same  time,  the  whole  significance  of  the  conception 
of  final  cause  is  that  it  is  in  itself  an  efficient  cause. 
When  an  occurrence  is  explained  in  terms  of  the  end 
which  a  conscious  and  thinking  being  has  in  view,  the 
end  or  final  cause  would  appear  to  be  functioning  as 
efficient  cause ;  but  it  must  be  borne  in  mind  that  the 
mere  idea  of  an  end  can  only  constitute  an  efficient 
cause  of  the  actualisation  of  that  end  in  so  far  as  it  in- 
volves an  act  of  will  which,  in  my  analysis,  constitutes  a 
crucial  occurrence  within  the  psychical  processes  which 
take  place  in  the  course  of  time. 

§  8.  Up  to  this  point  I  have  sketched  the  logical  con- 
ceptions employed  in  a  general  philosophical  account 
of  psychical  processes,  and  have  therefore  only  raised 
problems  open  to  philosophical  as  distinct  from  strictly 
psychological  controversy.  A  specifically  psychological 
explanation  of  the  way  in  which  feeling  and  desire  enter 


.4 


^. 


|i      *■ 


M» 


TRANSEUNT  AND  IMMANENT  CAUSALITY        141 

along  with  cognition  or  thought  as  jointly  determining 
the  critical  act  called  volition,  is  given  elsewhere  in  this 
volume ;  and  at  this  point  I  have  merely  put  forward 
a  brief  analysis  of  an  act  of  volition  as  guided  by  cog- 
nition and  motived  by  conation,  as  the  most  important 
illustration  of  purely  immanent  causality. 

In  further  explication  of  the  notions  of  transeunt  and 
immanent  causality  in  science,  it  must  be  pointed  out 
that  when  the  action  of  one  continuant  x  upon  another 
continuant  y  exhibits  transeunt  causality,  the  mode  in 
which  the  states  of  y  are  thus  causally  determined 
cannot  be  regarded  as  dependent  merely  upon  its  rela- 
tion to  X,  since  conjointly  with  this  relation,  the  mode 
depends  upon  some  character  peculiar  to  y  itself.  If 
this  specific  character  oiy  be  called  immanent,  we  have 
an  illustration  of  the  way  in  which  along  with  transeunt 
causality  an  immanent  factor  enters.  But,  when  we  con- 
ceive of  a  plurality  of  continuants  as  in  some  way  consti- 
tuting a  single  continuant,  then,  although  certain  causal 
processes  are  correctly  conceived  as  immanent  rela- 
tively to  this  unitary  whole,  yet  they  must  be  conceived 
as  transeunt  relatively  to  the  several  constituents.  The 
transeunt  interactions  between  the  several  constituents 
may  or  may  not  be  known  at  certain  stages  of  scientific 
development,  and  when  unknown  we  are  limited  to 
conceiving  the  processes  as  immanent  within  the  com- 
posite whole,  while  the  further  theoretically  possible 
knowledge  which  would  resolve  the  immanent  causality 
into  transeunt  processes  amongst  the  constituents,  would 
not  upset  our  previous  application  of  immanency,  but 
would  represent  a  more  penetrating  and  ultimate  know- 
ledge of  the  facts.    An  example  of  this  alternate  way 


142 


CHAPTER  IX 


of  representing  causal  processes  has  already  been 
given.  Perhaps  the  principle  called  the  conservation 
of  energy  would  illustrate  the  matter  as  well  as  any 
other.  This  principle  conceives  of  a  system  of  particles 
or  bodies  acting  on  one  another,  sometimes  in  highly 
complicated  modes,  preserving  throughout  all  its  changes 
a  constant  quantum  called  energy ;  and  thus  the  formula 
of  conservation  of  energy  represents  immanent  causality, 
inasmuch  as  it  regards  the  system  as  a  whole,  and  as 
not  transeuntly  operated  upon  from  without.  At  the 
same  time,  detailed  knowledge  of  the  forms  of  energy 
which  change  in  the  causal  process,  may  be  conceived 
under  transeunt  causality  as  operating  amongst  the 
particles  of  the  system.  This  illustration  from  physics 
may  be  compared  with  the  economic  conception  of  a 
society  of  persons.  Thus  the  formula  according  to  which 
prices  of  commodities  are  maintained  unchanged  so  far 
as  the  community  is  not  transeuntly  operated  upon  by 
other  communities,  is  analogous  to  the  conservation  of 
energy;  but  the  further  analysis  of  the  processes  of 
exchange  and  contract  between  person  and  person 
presents  the  facts  more  ultimately  and  more  exactly  as 
involving  transeunt  relations  between  the  persons;  just 
as  in  physics  processes  immanent  to  the  whole  are  more 
profoundly  defined  in  terms  of  transeunt  causality  as 
regards  its  parts. 


i\ 


-     r- 


^^ 


t 


I'   U 


I    ■« 


.'n' 


^  i 


«'      "•I 


4 


-        A 


CHAPTER  X 

CONVERGENT  AND  DIVERGENT  CAUSALITY 

§  I.  The  whole  topic  of  causal  determination  may  be 
approached  from  a  different  point  of  view  by  considering 
the  complex  relations  ofinterdependence  amongst  factors 
of  events  such  as  the  terms  cause  and  effect  are  fami- 
liarly used  to  describe.  It  will  simplify  the  exposition 
of  this  aspect  of  the  problem  to  introduce  a  little  elemen- 
tary symbolism,  and  throughout  this  chapter  the  reader 
is  asked  to  bear  in  mind  the  following  diagrams : 


//^^ 


a 


m 


If  both  a  change  in  a  alone  and  a  change  in  b  alone 
would  entail  or  point  to  a  change  in  /,  where  a  and  b  are 
cause-characters  and  /  an  effect-character,  then  we 
shall  speak  of  the  convergence  of  the  cause-characters  a 
and  b  towards/.  In  the  same  way,  if  both  a  change  in 
k  alone  and  a  change  in  /  alone  would  entail  or  point  to 
a  change  in  /,  where  k  and  /  are  considered  as  effect- 
characters  and  y  a  cause-character,  then  we  shall  speak 
of  the  divergence  of  the  effect-characters  k  and  /  from  f. 


144 


CHAPTER  X 


-T' 


Now,  when  a  and  b  converge  towards  /,  it  will  also  be 
the  case  with  respect  to  effect-characters  other  than  / 
— say:r  and  jy — that  a  change  in  a  alone  would  entail  a 
change  in  x  without  a  change  in  y,  and  a  change  in  b 
alone  would  entail  a  change  in  y  without  a  change  in  x. 
Processes  such  as  these,  from  a  to  x  and  from  b  to  y, 
may  be  said  to  be  parallel  to  one  another.  In  the  same 
way,  when  the  processes  from  f  to  k  and  f  to  I  are 
diverging,  it  will  also  be  the  case  with  respect  to  cause- 
characters  other  than  / — say  m  and  n — that  a  change 
in  k  alone  would  point  to  a  change  in  m  without  a 
change  in  n,  and  a  change  in  /  alone  would  point  to  a 
change  in  n  without  a  change  in  m.  Again  then, 
processes  such  as  that  from  mto  k  and  from  n  to  I  may 
be  said  to  be  parallel  to  one  another.  The  additional 
characters  a,^,K,X  are  introduced  in  the  above  diagrams 
in  order  to  exhibit  more  fully  the  significance  of  parallel, 
converging  and  diverging  processes. 

In  these  diagrams  the  horizontal  lines  from  left  to 
right  represent  the  course  of  time  from  before  to  after, 
so  that  any  vertical  line  that  may  be  imagined  represents 
simultaneity.  On  the  other  hand,  the  arrows,  which  are 
sometimes  directed  rightwards  and  sometimes  leftwards, 
indicate — not  the  temporal  opposition  of  before  and 
after — but  the  inferential  opposition  between  implying 
and  implied.  In  speaking  of  the  temporal  process  from 
ab  to  /  as  converging,  we  mean  that  when  a  and  b  are 
jointly  manifested,  they  both  play  a  part  in  determining 
for  us  what  value/  of  P  will  be  manifested.  This  con- 
verging process  is  represented  as  preceded  by  parallel 
processes  aa  and  ^b,  while  it  will  be  observed  that  the 
parallel  processes  ax  and  by  are  contemporaneous  with 


4»    U 


"  > 


I 


.  h 


^'     V 


I 


t^ 


^(1 


^^1 


/' 


I 
i 


tr  ^ 


;«'    y 


'o^       ^ 


*    > 


CONVERGENT  AND  DIVERGENT  CAUSALITY     145 

the   converging  process  from  ab  to  p.     Similarly   in 
speaking  of  the  temporal  process  from  /  to /^/ as  diverg- 
ing, we  mean  that  when  k  and  /  are  jointly  manifested, 
they  both  play  a  part  in  determining  for  us  the  value /" 
of  K     The  diverging  process  is  succeeded  by  parallel 
processes   ^k,  A,  while   again   the  parallel  processes 
mk,  nl,  are  contemporaneous  with  the  diverging  pro- 
cesses to  kl  from  /     In  the  interpretation  of  these 
diagrams,   it   is    important  not   to  be  misled  by  the 
irrelevant  spatial  relations  which   inevitably  enter   in 
diagrammatic  representations ;  for  example,  the  rela- 
tions of  parallelism  might  falsely  suggest  such  kinematic 
notions  as  two  particles  moving  contemporaneously  in 
parallel  directions  from  the  points  a,  )8,  to  the  points 
a  and  b,  and  then  converging  towards  the  point  /. 
A  special  application  of  these  diagrams  might  of  course 
be  made  to  such  a  kinematic  problem ;  but  if  it  were,  the 
full  significance  of  the  diagram  could  only  be  exhibited 
by  representing  the  two  particles,  whose  courses  are 
aa  and  ^b  respectively,  not  as  moving  in  parallel  lines, 
but  as  colliding  at  the  moment  represented  by  a  or  b, 
this  collision  altering  the  direction  of  the  particles  and 
accounting  for  the  state  of  things  at  the  moment  repre- 
sented by/,  which  closely  follows  the  moment  a  or  b. 
The  parallelism  of  the  lines  in  the  diagram  represents 
the  causal  or  determinative  independence  of  the  move- 
ments of  the  particles  prior  to  their  collision,  and  not 
their  spatial  directions ;  for  in  fact  they  must  have  been 
spatially  converging  in  order  to  lead  to  the  collision. 
Another   possible  interpretation    of  the  metaphor  of 
parallelism  which  it  is  important  to  avoid  in  the  use 
of  these  diagrams  is  that  which  applies  to  the  case  of 


J  L  III 


10 


146 


CHAPTER  X 


r 


psycho-physiological  parallelism.  There  a  knowledge 
of  the  physiological  processes  enables  us  to  infer  the 
contemporaneous  psychical  processes,  and  conversely  ; 
so  that  parallelism,  meaning  here  epistemic  determina- 
tion, would  be  represented  by  two  oppositely  directed 
arrows.  But  in  the  case  which  the  diagram  repre- 
sents, and  which  might  be  illustrated  by  two  billiard 
balls,  the  motion  of  one  ball  before  collision  with 
the  other  would  inferentially  determine  no  knowledge 
whatever  of  the  preceding  velocity  or  direction  of  the 
other.  Parallelists  who  tried  to  combine  the  notions  of 
determinative  dependence  and  causal  independence 
would  have  to  revise  the  entire  logical  account  of 
causality  and  its  connection  with  epistemic  determina- 
tion. 

§  2.    It  will  give  additional  significance  to  our  dia- 
grams to  further  elaborate  the  kinematic  problem.    The 
straight  part  aa  of  the  crooked  line  aap,  might  stand  for 
a  movement  of  uniform  velocity  and  constant  direction, 
or  rather  for  the  constant  retardation  due  say  to  the 
friction  etc.  of  a  ball  moving  on  a  billiard  table.    Then 
the  change  from  aa  to  ap  would  represent  the  change 
of  acceleration  or  retardation  which  takes  place  at  the 
moment   of  geometrical   contact  with    a   second  ball 
whose  previous  course  is  represented  by  ^6  and  whose 
changed  acceleration  is  shown  by  dp.    Furthermore  the 
junction  of  p  with  /  might  represent  the  process  of 
contraction  and  re-expansion  due  to  the  elasticities  of 
the  balls.    And,  lastly,  kK  and  /X  would  exhibit  their 
subsequent  rectilinear  and  causally  independent  move- 
ments. 

To  show  that  these  diagrams  are  adaptable  to  very 


«> 


^  ^' 


I 


w  „^ 


.J 

•■ll 


^1 


II 


/ 


i,4-'      ^ 


-      ;^ 


CONVERGENT  AND  DIVERGENT  CAUSALITY     147 

different   kinds   of  phenomena,   let   us    now   turn   to 
chemistry,  and  suppose  a  to  represent  the  continuously 
manifested  properties  of  a  sample  of  oxygen,   and  b 
those   of  a   sample   of  hydrogen.     These  properties 
would  continue  to  be  manifested  without  change  until 
some  environmental  condition  brought  the  two  sub- 
stances into  a  special  spatial  and  physical  connection.    If 
a  and  b  represent  the  moment  at  which  this  connection 
takes  place,  then  at  the  moment  ab  there  is  initiated  a 
process  of  chemical  combination  which  leads  to  the  mani- 
festation  of  new  properties— viz.  the  properties  of  water 
— symbolised  by/.    The  arrow  from  ab  to/  represents 
the  fact  that  our  knowledge  that  oxygen  and  hydrogen 
are    the   elements   concerned  determines   for   us   the 
knowledge  that  the  properties  of  water  will  be  mani- 
fested   in    the    combination;     these    properties   being 
jointly  determined  by  the  definition  of  the  elements 
and  their  spatial  relations.  As,  in  the  kinematic  illustra- 
tion, so  here  an  appearance  of  discontinuity  or  abrupt- 
ness is  presented  at  the  moment  when  the  oxygen  and 
hydrogen  combine,  and  are  replaced  by  the  properties 
of  water.    Further  ax  and  by  may  be  taken  to  repre- 
sent the  continuance  unchanged  of  the  weights  of  the 
oxygen  and  the  hydrogen  that  are  combining;  these 
unchanged  and  independent  values  continuing  to   be 
manifested  contemporaneously  with  the  process  which 
ends  in  the  chemical   combination.    In   this  way  the 
illustration  from  chemistry  affords  an  example  of  the 
universal  principle  that  along  with  any  converging  pro- 
cess of  causality,  there  are   always  contemporaneous 
parallel  processes — the  words  parallel  and  converging 
being  used  metaphorically  to   stand   respectively  for 


r 


10 — 2 


148 


CHAPTER  X 


causal  independence  and  joint  dependence.  By  bringing 
into  line  these  examples  from  chemistry  and  dynamics 
we  have  partially  shown  in  what  respects  chemical  com- 
bination and  dynamical  composition  agree,  and  in  what 
respects  they  differ.  The  complete  change  of  motion  of 
the  two  colliding  balls  corresponds  to  the  change  from 
the  manifestation  of  the  chemical  properties  of  oxygen 
and  hydrogen  to  that  of  the  properties  of  water.  And 
the  continuance  of  the  weights  of  the  oxygen  and 
hydrogen  unchanged  during  the  process  of  combination 
corresponds  to  our  conception  of  the  continuance  of  the 
masses  and  resultant  momenta  of  the  two  balls  during 
the  process  of  collision. 

§  3.  The  same  diagrams  serve  to  illustrate  the  causal 
connections  between  sensation  and  physical  stimulus. 
We  will  suppose  that  the  moment  ^  or  ^  is  that  at  which 
the  subject  becomes  aware  of  two  sensations — say  of  the 
colours  red  and  blue  present  simultaneously  apart  in  the 
field  of  vision.  The  physical  processes  preceding  these 
sensations  are  represented  by  aa  and  ^b,  parallelism 
of  the  lines  standing  for  the  presumed  causal  indepen- 
dence of  these  two  physical  processes.  Now,  when  the 
sensations  a  and  b  happen  to  be  incidents  in  the  experi- 
ence of  the  same  percipient,  there  will  be  consequences 
which  would  not  be  entailed  if  these  sensations  were  ex- 
perienced by  different  percipients ;  these  consequences, 
which  we  will  suppose  to  be  the  apprehension  of  a 
relation  of  difference  between  the  two  sensations,  will 
be  represented  by  the  converging  process  from  ab  to  /. 
The  arrow  from  ab  to  /  illustrates  the  fact  that  the 
apprehension  of  the  relation  of  difference  will  be  deter- 
mined by  the  apprehensions  ab  jointly.    The  contem- 


A 


X 


f. 


1 


.r 


^V  ij 


^» 


<  I 


^>  --> 


7^ 

I 

A. 


CONVERGENT  AND  DIVERGENT  CAUSALITY     149 

poraneous  processes  ax,  by,  might  serve  to  illustrate  the 
continuance  unchanged  of  the  apprehensions  a  and  b, 
while  the  apprehension  of  their  relation  of  difference  is 
being  evolved.  But  there  are  many  other  ways  in 
which  the  diagram  could  be  interpreted  to  illustrate  the 
psycho-physical  process  of  sensory  stimulation.  For 
instance,  if  a  and  b  stood  for  the  neural  processes  occur- 
ring contemporaneously  with  the  sensations,  these  would 
continue  to  pursue  their  course,  in  some  respects  at 
least,  unaffected  by  the  percipient's  cognitive  processes 
of  comparison  and  so  on  ;  and  in  this  case  the  more  or 
less  parallel  processes  ax  and  by  would  be  contem- 
poraneous with  the  converging  process  from  ab  to  p. 

The  psycho-physical  illustration  may  be  carried  further 
by  supposing  the  points  /  and/"  to  be  joined  up;  this 
enables  us  to  treat  a  more  complex  problem.  Thus,  if 
a  and  b  stand  for  different  simultaneous  occurrences 
whose  relations  or  connections  are  apprehended  in  a 
synthetic  cognitive  act  (symbolised  by/),  the  nature  of 
this  perception  is  causally  determined  by  the  nature  of 
a  and  b  jointly ;  hence  the  arrow  passing  from  ab  to  p. 
Next  taking/  as  cause,  the  effects  which  follow  will  be 
other  than  those  which  could  have  been  predicted  from 
a  knowledge  of  the  separate  processes  aa  and  fib ;  these 
latter  consequences,  which  occur  independently  of  /, 
will  be  symbolised  by  the  lines  ax,  by,  continued  inde- 
finitely in  straight  or  converging  directions,  of  which 
the  course  may  be  said  to  be  parallel  to  the  mental 
processes  and  their  consequences — the  word  parallel 
being  used  metaphorically  to  signify  determinative  in- 
dependence. For  instance,  such  phenomena  as  light  or 
heat  will  engender  various  physical  consequences  in  the 


150 


CHAPTER  X 


^^1 


outer  world  concurrently  with  the  mental  processes  and 
purposes  of  any  individual  percipient ;  and  these  physi- 
cal consequences  will,  in  most  of  their  aspects,  be 
independent  of  psychical  process  until  some  new  con- 
verging process,  involving  what  we  may  call  metaphori- 
cally another  collision  between  mind  and  matter,  takes 
place.  Meantime,  let  us  consider  in  further  detail  the 
effects  following  upon  p,  which  represented  the  con- 
vergence of  physical  causes  to  a  psychical  effect,  these 
effects  being  represented  by  divergent  processes  in  which 
the  causality  is  from  the  psychical  to  the  physical.  Let 
us  suppose  that  the  perception/  develops,  owing  to  such 
conditions  as  the  percipient  s  character  and  past  experi- 
ence, through  processes  of  cognitive  and  conative  deli- 
beration, into  a  fiat  of  the  will  (symbolised  by/).  The 
causal  process  of  inner  deliberation  is  represented  by  a 
line  which  may  be  supposed  to  join  p  to/.  Then  giving 
to  the  effects  of/"  the  same  kind  of  complexity  that  we 
have  attributed  to  the  causes  of/,  k  and  /may  be  taken 
to  represent  the  diverging  manifestations  of  the  fiat/. 
The  arrow  pointing  from  k/  to  /  indicates  that  the 
observer  can  infer  from  the  joint  manifestation  of  k  and 
/  the  character  of  the  fiat  /  which  caused  this  mani- 
festation. If  then  k  and  /  represent  those  phases  in  the 
causal  process  over  which  the  experient  has  no  longer 
any  direct  control,  k  will  develop  causally  into  k,  and 
/  into  X,  along  independent  lines,  such  that  from  k  alone 
we  could  infer  k  as  its  cause,  and  from  X  alone  we  could 
infer  /  as  its  cause.  Thus  the  two  parts  of  the  dia- 
gram are  joined  up,  and  it  is  seen  how  the  two  inde- 
pendent causal  processes  aa  and  ^b  may  issue — through 
the  intermediary  processes  from  /  to  /—into  the  two 


<f 


k> 


vW 


"^ 


J 


r^ 


CONVERGENT  AND  DIVERGENT  CAUSALITY      151 

independent  causal  processes  kK  and  /X.  Now  know- 
ledge of  the  laws  of  causal  determination  according  to 
which  a  evolves  into  a,  and  ^  into  b,  would  not  by  it- 
self enable  us  to  derive  the  subsequent  processes  Kk  and 
f  /X.  Though  all  these  four  processes  be  taken  to  exhibit 
I  the  laws  according  to  which  physical  phenomena  are 
^regulated,  no  mere  physical  law  will  account  for  kK  and 
l\  as  consequences  of  aa  and  ^b.  To  explain  these 
physical  consequents  of  physical  antecedents,  we  must 
interpolate  a  converging,  an  internal,  and  a  diverging 
process  of  causal  determination  whose  sphere  of  opera- 
tion is  the  inner  consciousness  of  an  individual  experient. 
\The  joint  diagram  may  be  shortly  said  to  represent  the 
alternate  action  of  matter  upon  mind  and  mind  upon 
matter. 

With  regard  to  inference  in  the  case  of  divergent 
causal  process,  while  the  distinct  lines  fk,  fl  indicate 
that  the  process  may  be  analysed  into  two  or  more 
distinct  part  processes,  the  single  arrow  pointing  back- 
wards from  kl  to  /  indicates  that  in  general  we  can 
infer  the  determinate  value/,  only  from  a  knowledge  of 
both  k  and  /,  and  not  from  k  alone  nor  from  /  alone. 
The  symbols  may,  however,  represent  an  analysis  from 
which /could  in  certain  cases  be  inferred  either  from  k 
alone  or  from  /alone.  To  give  an  instance  of  inference 
from  joint  factors  forwards  and  backwards,  we  may  pur- 
sue the  course  of  two  billiard  balls,  forwards  from  aa  and 
^b  to  Kk  and  /X,  taking/  to  represent  the  forces  of  com- 
pression on  both  the  balls,  and  /  the  forces  of  expansion 
on  both  the  balls;  then/  would  contain  the  conditions 
from  which  we  could  infer  backwards  both  a  and  b,  and 
/would  contain  the  conditions  from  which  we  could 


M 


152 


CHAPTER  X 


^ 


infer  forwards  both  k  and  /.  Equally  well  we  could 
have  inferred  forwards  from  (xxi  and  ^b  the  moment  and 
position  at  which  the  balls  will  touch,  and  from  kK  and 
A  we  can  infer  backwards  the  moment  and  position  at 
which  they  have  touched.  But  we  cannot  infer  kK  from 
aa  alone,  nor  aa  from  kK  alone  without  taking  into 
consideration  the  movement  of  the  other  ball  which 
introduces  the  converging  and  diverging  processes. 
Illustrations  of  the  diverging  process  in  which  we  infer 
backwards  from  the  conjunction  of  two  or  more  effects, 
the  nature  of  the  cause,  are  well  furnished  by  the  case 
of  symptoms.  Thus  in  medical  diagnosis  it  is  often 
impossible  to  infer  the  nature  of  a  specific  disease  from 
any  of  the  symptoms  separately,  and  it  is  therefore 
necessary  to  join  different  symptoms  in  order  to  infer 
their  cause.  Similarly  the  effects  of  different  emotions 
such  as  anger  and  fear,  as  manifested  in  bodily  dis- 
turbances, partially  agree  and  partially  differ ;  hence  a 
number  of  factors  would  have  to  be  noted  in  order  to 
infer  in  any  given  case  whether  the  cause  of  the  bodily 
disturbances  was  fear  or  anger.  Purposive  action  af- 
fords a  peculiarly  interesting  example  of  our  analysis 
of  causality  into  converging  and  diverging  processes. 
Such  action  may  in  general  be  defined  as  involving  a 
divergent  process  issuing  from  a  thought  of  an  end, 
followed  by  a  convergent  process  in  the  outer  environ- 
ment in  which  this  same  end,  previously  represented  in 
thought,  is  actualised  in  fact. 

§  4.  We  will  now  consider  certain  more  complicated 
cases  of  causal  process  which  exhibit  convergent,  diver- 
gent and  parallel  processes  contemporaneously.  For 
illustration  we  will  take  two  particles  whose  movements 


^» 


7 

t 


V 


4  ^ 


>:^ 


I 


n 


'4 


CONVERGENT  AND  DIVERGENT  CAUSALITY     153 

are  determined  under  some  such  law  as  that  of  gravity. 
Let  a  be  one  particle,  and  let/^  and/'^  be  two  positions 
successively  occupied  by  a  at  two  moments  separated  by 
an  assigned  interval  of  time.  We  cannot  infer /'^  from 
pa  alone,  but  only  from  knowledge  oi p^  jointly  with  the 
change  of  motion  which  a  is  undergoing  when  at  posi- 
tion p\  if  this  change  of  motion  be  symbolised  by  c^y 
we  may  then  speak  oi  p' ^  as  determined  jointly  by/^ 
and  c^.  Adapting  our  previous  diagram  to  this  relation 
of  causality,  we  have  the  following : 


Pa 


P'a 


where  p^,  c^.p'^  take  the  place  of  a,  b,  p  respectively, 
and  the  figure  represents  a  converging  process.  Before 
introducing  a  second  particle  b,  we  will  simplify  the 
above  diagram  by  bringing  c^  close  up  to  p^,  under- 
standing by  this  juxtaposition  literal  simultaneity,  and 
then  join  /^,  c^  to  p' ^  by  the  horizontal  arrow.  The 
motion  of  particle  b  is  similarly  represented  by  symbols 
in  the  second  diagram.  Now  when  a  is  at  position  /«, 
and  b  is  simultaneously  at  position  p^y  the  distance 
between/^  and/^  is  a  determining  condition  from  which 
we  can  infer  the  change  of  motion  of  both  a  and  b,  under 
the  law  which  we  have  assumed  to  be  that  of  gravity  : 
this  relation  of  distance,  therefore,  stands  as  a  cause- 
condition  diverging  into  the  two  effects  c^  and  Cb.  The 
process  is  exhibited  in  the  following  diagram,  which  also 
illustrates  subsequent  positions  of  the  particles  subject 
to  the  same  type  of  causality.  Here,  at  the  first  moment, 
the  distance  d  between/^  and/^  determines  divergently 
the  changes  of  motion  c^  and  c^,  while  the  position/^ 
and  the  change  c^  determine  convergently  the  position 


154 


CHAPTER  X 


*< 


p\  ;  similarly  the  position /^  and  the  change  Cj,  determine 
convergently  the  position  /V  It  is  also  important  to  note 
that  the  converging  process /^^^  to/'^  is  determinatively 
parallel  to,  that  is  independent  of,  the  converging  process 
ixovcL  pifi,  to/V  The  same  relations  are  exhibited  at 
the  next  moment  considered  in  relation  to  the  third 
moment,  where  the  dashes  serve  to  distinguish  the 


Pa  C^ 


several  moments.  Between  each  of  the  moments 
separated  in  the  diagram,  we  must  suppose  an  indefinite 
number  of  the  same  configurations  following  one  another 
continuously.  And  since,  at  any  instance,  the  distance 
^  is  a  causal  factor  common  to  the  movements  of  a  and 
of  ^,  the  movements  from/^  to/'^  and  from/^  to/'^  are 
not  properly  called  parallel  in  the  determinative  sense, 
when  an  appreciable  interval  of  time  has  elapsed.  It 
must  also  be  remembered  that  the  positions  and  juxta- 
posed changes  of  motion  are  to  be  conceived  as  literally 
simultaneous  and  not  as  continuously  successive.  We 
have  spoken  oip^  and  c^  as  jointly  determining/'^;  we 
may  equally  speak  of/^  and/'^  as  jointly  determining 
c^\  and  this  illustrates  the  commutative  principle  for 
what  has  been  called  a  prime  dependency.  The  three 
values  pa,  Cay  p'a  Hiay  be  said  to  constitute  a  kinematic 
prime  dependency :  it  is  actually  by  the  observation  of 
pa  and/'^  that  we  infer  ^^,  while  we  regard/^  and  Ca  as 
causally  determining/'^.    Thus  a  knowledge  oipa.p'a 


k\ 


4) 


A 


<   if 


Ay 


^p 


^ 


CONVERGENT  AND  DIVERGENT  CAUSALITY     155 

and/^,  /'^,  could  take  the  place  of  the  knowledge/^,  c^, 
SLTidpiy  Ciy  in  determining  the  whole  course  of  the  action 
both  backwards  and  forwards.  The  diagram  could  then 
be  simplified  by  omitting  the  symbols  for  change  of 
motion  on  a  principle  analogous  to  the  triangle  of  forces, 
so  that  a  single  arrow  from  d  to  /'^  will  replace  the  two 
arrows  from  d  to  c^  and  c^  top\,  thus  : 


To  indicate  the  continuity  of  the  process,  we  may  still 
further  condense  the  diagram.  In  the  former  of  these 
two  diagrams  pa  and  d  determine      /,  / '        X' 

convergently  /'^ ;  while  p^  and  d 
convergently  determine/'^,  these 
two  processes  being  themselves 

divergent.  In  the  latter  of  the  two      

diagrams  we  represent/^  and /^  as     ^*  P*       Pi 

convergently  determining  both/'^  and/'^,  while /'«  and 
p'j,  are  divergently  determined  by/^  and/^.  A  slightly 
different  interpretation  of  the  symbols  will  elucidate  this 
apparent  contradiction.  If  the  symbols  /,  instead  of 
representing  the  mere  geometric  notion  of  position,  be 
interpreted  kinetically  to  include  position  and  deter- 
minate motor  tendency,  the  relation  doi  bloa  will  then 
be  conceived  as  a  causal  condition  modifying  the  motor 
tendency  and  thus  effecting  an  actual  motion  other  than 
that  which  the  tendency  by  itself  would  have  produced. 


n 


156 


CHAPTER  X 


^■ 


The  cause  conceived  in  this  way  has  no  effect  peculiar 
to  itself,  but  modifies  what  would  otherwise  have  been 
the  determinate  effect.  The  phenomenon  b,  when  in  the 
relation  d  to  a,  which  thus  modifies  the  process  of  a,  is 
conceived  as  agent  relatively  to  a,  and  a  is  conceived 
as  patient  relatively  X.o  b.  In  philosophical  terminology 
we  speak  of  the  unmodified  process  of  a  as  illustrating 
immanent  causality,  and  the  modifying  influence  of  b 
upon  a  as  illustrating  transeunt  causality.  Apart  from 
these  disputable  terms,  the  consideration  with  which  we 
are  here  concerned  is  that  in  order  to  define  the  nature 
of  the  effect  which  the  relation  d  oi  b  has  upon  a,  it  is 
necessary  to  introduce  reference  to  the  trend  or  motor 
tendency  which  a  is  manifesting  at  the  moment  when 
b  influences  it.  This  serves  to  illustrate  the  point  that 
the  idea  of  change  is  complex,  and  needs  to  be  carefully 
examined:  it  does  not  mean  simply  a  difference  in 
the  state  of  a  at  one  moment  as  compared  with  its 
state  at  a  subsequent  moment,  but  it  means  a  difference 
between  the  state  into  which  a  passes  under  the  opera- 
tion of  an  external  causal  agency,  such  as  its  determining 
relation  d  to  <5,  as  compared  with  the  state  into  which 
a  would  have  passed  by  its  own  agency. 

§  5.  This  complex  form  of  causality  may  be  illustrated 
from  psycho-physical  process  as  well  as  from  dynamics. 
In  this  case  the  symbol  a  in  the  condensed  diagram 
will  stand  for  the  mental  side  of  such  a  process,  and  the 
symbol  b  for  the  physical  side ;  the  letters  p  representing 
not  statically  defined  states,  but  motor  trends.  Thus  if 
p^  represents  the  course  which  a  sensation  process  is 
taking  at  any  moment  independently  of  any  physical 
process  such  as  b,  the  physical  stimulus/^,  as  soon  as 


r 


<s 


vWr 


V 


I 

4' 


CONVERGENT  AND  DIVERGENT  CAUSALITY      157 

it  begins  to  operate,  will  affect  this  sensational  trend, 
and  determine  it  in  the  form  p\,  which  is  a  modifica- 
tion of  what  p^  would  have  become  apart  from  the 
stimulus.  Now,  if  the  subject  is  what  may  be  described 
as  inactive  with  regard  to  the  further  course  of  his  sen- 
sations, the  arrow  in  our  diagram  will  be  drawn  only 
from  the  line  of  b  to  the  line  of  a,  and  the  arrows  in  the 
opposite  direction  may  be  omitted.  The  diagram  would 
then  represent  a  state  in  which  the  sequence  of  sensa- 
tions was  wholly  determined  by  the  course  which  the 
physical  or  physiological  processes  assume  under  purely 
physical  laws,  and  where  there  was  no  reaction  from  the 
side  of  the  psychical  to  the  side  of  the  physical.  But 
now  let  us  suppose  that  the  subject  is  active  and  takes 
a  share  in  determining  the  course  of  his  sensations.  It 
must  be  admitted  that  such  active  determination  by  the 
subject  is  not  a  direct  causal  determinant;  and  the  facts 
are  illustrated  by  the  diagram  with  all  the  arrows  inserted. 
Thus  we  shall  define  p^  not  as  a  mere  passively  received 
sensation,  but  as  a  cognition,  having  in  it  an  element 
determined  by  the  nature  of  the  stimulus/^,  and  besides 
this,  other  related  cognitive  elements  more  or  less  com- 
plicated according  to  the  degree  of  intelligence  of  the 
supposed  subject.  The  arrow,  therefore,  from  p^  to  /'^ 
indicates  that  /^,  defined — not  as  a  mere  sensation 
— but  as  a  cognition,  causally  determines/'^  in  the  same 
way  as  pi  was  previously  shown  to  determine/'^;  that 
\s,pa  does  not  bring /'^  into  existence,  but  it  determines 
the  actual  form  assumed  by  the  physiological  process  by  in 
the  sense  of  modifying  the  form  pj,  would  have  taken, 
apart  from  the  active  determination  of  this  cognition.  The 
whole  process  is  descriptively  condensed  in  the  phrase 


158 


CHAPTER  X 


that  the  physiological  course  and  the  sensational  course 
reciprocally  determine  one  another;  neither  would  be 
what  it  actually  is,  if  the  influence  of  the  other  had  been 
non-determinative.  This  condensed  description,  how- 
ever, is  more  accurately  analysed  into  an  alternate 
process  from  the  side  of  mentality  to  the  physiological 
and  reversely  from  the  physiological  to  the  side  of 
mentality.  The  mentality  involved  is  not  purely  passive 
sensation,  but  actively  determinative  cognition,  involving 
(at  least)  what  psychologists  call  attention ;  and  in  cases 
of  a  higher  level  of  intelligence,  more  or  less  co-ordinated 
purpose.  The  process  indeed,  which  the  subject  cog- 
nises, is  itself  mental,  and  must  not  be  confused  with 
the  course  of  the  physiological  changes  themselves,  of 
which  he  is  wholly  unaware;  his  attention  is  actually 
directed  to  the  changes  in  the  sensational  experience 
of  which  he  is  retrospectively  and  more  or  less  prospec- 
tively aware. 

The  condensed  diagram  interpreted  so  far  to  apply 
to  the  causal  interrelations  between  a  merely  physio- 
logical process  on  the  one  side  and  active  mentality 
on  the  other,  can  be  used  to  illustrate  a  wider  range 
of  interaction  between  mind  and  matter,  which  shall 
include  operations  on  the  external  environment.  In 
this  application  the  line  b  no  longer  represents  purely 
physiological  processes,  but  includes  processes  in  the 
external  physical  world.  Here  again  the  important 
consideration  is  that  the  purposive  thought /^  does  not 
bring  into  existence  the  physical  phenomenon  p\,  but 
it  determines  the  phase  of  b  to  be  otherwise  than  what 
p  would  have  become  under  the  determination  of  purely 
physical  causality.    Of  course  the  mode  in  which  the 


M 


i 


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(■'. 


;j 


(> 


St.. 


i^\ 


f 


4k.» 


4 


I* 


\ 


CONVERGENT  AND  DIVERGENT  CAUSALITY      159 

course  of  p^  passes  into  the  phase  p'f,  is  not  adequately 
represented  by  a  single  straight  line,  and  in  the  diagram 
a  very  complicated  process  is  artificially  condensed.  I  n 
fact  there  will  be  sections  of  the  physical  process  that 
are  left — uninfluenced  by  the  subject — to  follow  a  course 
determined  by  purely  physical  causality;  and  to  represent 
such  sections,  the  arrows  from  the  side  of  mentality  to 
the  physical  side  should  properly  be  omitted.  Changes 
of  this  kind  may  be  observed  by  the  subject,  and  his 
observation  of  the  phase  into  which  the  process  has 
passed  may  determine  him  to  initiate  a  new  interfering 
or  controlling  operation  which  will  again  modify  the 
further  course  of  the  physical  process.  The  moment 
when  this  observation  occurs  will  be  marked  by  an  arrow 
from  the  physical  side  to  the  side  of  mentality,  since  it 
is  the  nature  of  the  physical  occurrence  which  determines 
the  content  of  the  predicative  cognition  on  the  part  of 
the  observer.  In  its  turn,  this  cognition  will  operate  on 
the  other  variously  modifiable  inner  processes  of  the 
mind,  and  determine  a  corresponding  reaction,  modifying 
the  physiological  as  well  as  the  physical  course  of  things ; 
and  these  changes  will  be  marked  by  an  arrow  from  the 
side  of  mentality  to  that  of  the  external  and  physical. 
This  arrow  is  again  a  condensed  representation  of  con- 
verging process;  for  the  phase /'^  determined  from  the 
side  of  mentality,  is  due  jointly  to  the  just  preceding  obser- 
vation, taken  along  with  the  appreciation  oip\  as  a  phase 
in  the  progressively  attained  purpose,  as  well  as  the 
knowledge  of  the  activities  needed  for  furthering  this 
attainment. 

Finally  we  may  close  the  exhibition  of  the  entire 
purposive  process  by  a  set  of  lines  converging  upon 


i6o 


CHAPTER  X 


that  terminal  phase  of  actualised  experience  which 
denotes  the  realisation  of  the  end  corresponding  to  the 
thought  of  the  end  from  which  at  the  beginning  the 
initial  processes  diverged.  Thus,  the  scheme  of  pur- 
posive causality  begins  and  ends  as  a  phase  in  the  con- 
sciousness of  the  same  individual  thinker  or  actor;  while 
the  intermediate  or  instrumental  phases  are  incidents  in 
the  world  of  physical  phenomena,  some  of  which  are 
within  the  organism  and  nervous  system,  and  thus  in  the 
most  direct  causal  contact  v/ith  the  thinker  s  feelings, 
thoughts,  and  powers  of  causal  determination. 


\\ 


i4    > 


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4 


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>* 


I 


I 


i 


^.w 


r 


^ 


CHAPTER  XI 

TEMPORAL  AND  SPATIAL  RELATIONS 
INVOLVED  IN  CAUSALITY 

§  I.  The  general  discussion  of  connectional  determina- 
tion entails  consideration  of  the  spatio-temporal  relations 
amongst  phenomena  in  terms  of  which  occurrences  are 
represented  as  bound  together  in  a  unity  of  connection. 
Thus,  the  bare  formula  'adcde  determines/,*  where  the 
symbols  stand  for  the  characterising  adjectives  of  occur- 
rences, is  a  merely  abstract  expression  of  the  causal  prin- 
ciple, inasmuch  as  no  reference  is  explicitly  made  to  the 
spatio-temporal  nexus  (as  it  may  be  termed)  under  which 
the  manifestations  of  these  characters  take  place.  When 
the  event  characterised  as  adcde  is  said  connectionally 
to  determine  an  event  characterised  as  /,  these  events 
have  spatio-temporal  extension  as  also  spatio-temporal 
relations  one  with  another.    The  manifestations  a,  b,  c,  d,  e 
will  be  termed  occurrents  severally,  and  their  conjoint 
manifestation  will  be  termed  an  event.   These  occurrents 
are  several,  because  the  determinate  character  of  each 
comes  under  its  own  determinable  A,  B,  C,  D,  E  re- 
spectively.   On  the  other  hand,  events  are  several,  be- 
cause each  has  its  own  distinct  spatio-temporal  boun- 
dary.   The  extension  of  an  event  allows  us  to  speak  of 
any  event  as  containing  spatio-temporally  distinguish- 
able parts,  which  are  themselves  events.    On  the  other 
hand,  the  occurrents  a,  b,  c,  d,  e  are  not  parts  but  con- 
stituents of  the  event,  for  they  all  occupy  one  and  the 
same  spatio-temporal  position  defined  by  the  boundary 


J  LIII 


II 


l62 


CHAPTER  XI 


of  the  event.  Furthermore,  connectional  determination 
signifies  that  xh^ position  of  the  manifestation  of/  rela- 
tively to  that  of  abcde,  as  also  its  determinate  character 
p,  is  determined  jointly  by  the  characters  of  the  coin- 
cident manifestations  a,  b,  Cy  dy  e. 

So  far  we  have  treated  the  notion  of  nexus  as  con- 
cerned solely  with  spatio-temporal  relations.  But  the 
above  account  must  be  amplified  and,  in  a  sense,  par- 
tially amended  by  noting  that  every  occurrent  must  be 
referred  to  its  own  proper  continuant.  Thus,  to  the  con- 
trast between  occupying  the  same  or  different  positions 
must  be  added  that  between  being  referred  to  the  same 
or  to  different  continuants.  I  n  fact,  distinctions  of  position 
must  be  understood  metaphorically  to  extend  to  distinc- 
tions of  continuant-reference.  And,  for  similar  reasons, 
determinables  must  be  distinguished  according  as  their 
determinate  values  characterise  manifestations  referred 
to  this  or  to  that  continuant. 

§  2.  We  will  now  examine  the  general  notion  of 
Order.  Order  is  predicated  of  terms  which,  for  con- 
venience of  figurative  representation,  may  be  called 
Points,  and  when  Points  are  in  an  Order  the  collection 
is  called  a  Series.  Taking  any  three  points  whatever  in 
a  Series,  these  may  be  said  to  be  in  a  determined  order 
when  there  is  some  assignable  principle  according  to 
which  one  of  the  three  points  is  to  be  regarded  as 
'between'  the  other  two.  Thus  the  three  notions  of  a 
Series,  or  Order,  and  of  'betweenness,'  mutually  in- 
volve one  another.  These  remarks  apply  equally  to 
what  may  be  called  a  discrete  Series,  or  to  a  con- 
tinuous Series — two  types  of  Order  which  may  be 
distinguished  as  follows  :  a  continuous  series  is  such 


-i 


4 


V 


'•i.^ 


/ 


TEMPORAL  AND  SPATIAL  RELATIONS  163 

that  between  any  two  non-identical  Points,  there  is  a 
Point  non-identical  with  both.    The  series  of  integers 
i»  2y  3»  4)---»  on  the  other  hand,  illustrates  a  discrete 
series,  for,  between  the  two  integers  3  and  4  for  in- 
stance, there  is  no  integer  ;  again  the  dots  after  4  may 
also  be  taken  to  illustrate  a  discrete  series,  for  between 
the  first  and  second  there  is  no  dot,  as  also  between  the 
second  and  third,  and  so  on.     A  discrete  Series  is,  in 
fact,  always  figuratively  represented  by  dots  spatially 
separated  along  a  line ;  and  the  fact  that  it  is  natural  to 
name  these  points  by  the  ordinal  numbers  shows  that  a 
series  of  integers  most  naturally  illustrates  a  discrete 
Series.    Similarly  it  is  natural  figuratively  to  represent 
a  continuous  Series  by  a  line,  containing  points  such 
that  between  any  two  points  there  is  in  the  Series  a  point 
different  from  both.  When  a  line  is  regarded  as  the 
boundary  between  two  contiguous  areas  of  a  surface,  it 
enables  us  to  conceive  of  a  discrete  series  of  areas ;  thus 
we  can  count  one  by  one  a  series  of  contiguous  areas 
by  mentally  representing  the  lineal  boundary  common 
to  any  two;  but  in  such  case  the  entire  surface  is  to  be 
described  as  continuouSy  for  between  any  two  lineal 
boundaries,  there  is  in  this  surface  a  lineal  boundary 
different  from  both.   The  surface  itself  may  be  regarded 
as  the  boundary  dividing  a  region  into  a  discrete  series 
of  parts ;    but  again  in  this  case,  the  entire  region  is  to 
be  described  as  continuous ;  for  between  any  two  areal 
boundaries  within  the  region,  there  is  an  areal  boundary 
different  from  both.    We  are  thus  led  to  distinguish 
between    the  parts  of  a  whole,  and  the   boundaries 
between  these  parts.    The  parts  of  a  line  are  lines,  the 
parts  of  an  area  are  areas,  the  parts  of  a  region  are 

II— 2 


164 


CHAPTER  XI 


regions ;  but  the  boundary  between  contiguous  parts 
of  a  line  is  a  point,  and  the  boundary  between  con- 
tiguous parts  of  an  area  is  a  Hne,  and  the  boundary 
between  contiguous  parts  of  a  region  is  an  area  or  surface. 
The  parts  of  a  whole,  therefore,  are  homogeneous  with 
one  another  and  with  the  whole ;  but  as  we  noted  when 
contrasting  extensive  with  extensional  wholes  \  the 
boundaries  between  two  contiguous  parts  are  always 
of  one  lower  order  of  dimensions  than  the  parts. 

Our  illustrations  of  discreteness  and  continuity  have 
so  far  been  taken  solely  from  Space ;    but  the  notions 
are  equally  applicable  to  Time.   Thus  Time  is  conceived 
as  of  one  dimension,  and  is  composed  of  parts  which  are 
periods,  the  boundary  between  two  contiguous  periods 
being  called  an  instant.    A  period  of  time,  therefore, 
corresponds  to  a  line,  and  an  instant  corresponds  to 
a  point ;  the  period-parts  of  a  period  will  then  constitute 
a  discrete  Series,  and  the  instants — i.e.  the  boundaries 
between  two  contiguous  period-parts — will  constitute  a 
continuous  Series.    The  above  application  of  the  term 
*  discrete'  to  contiguous  parts  of  a  whole  might  be 
criticised  as  being  incompatible  with  its  original  appli- 
cation to  separated  points.     But  it  must  be  noted  that 
the   notion   of   discreteness   does    not   imply  factual 
separation,  but  only  separation  in  thought.     When  we 
think  of  a  boundary  between  two  contiguous  parts,  we 
are  mentally  separating  those  parts,  without  predicating 
any  factual  separation ;  in  this  sense  we  may  always  say 
that  a  whole  can  be  conceived  as  a  sum  of  its  discrete 
parts,  whether  the  whole  is  such  that  it  can  be  said  to 
contain  contiguous  parts,  i.e.  parts  having  a  common 

^  Part  II,  Chapter  VII,  §  8. 


TEMPORAL  AND  SPATIAL  RELATIONS 


165 


/. 


J 


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<   4 


,M.  <  ^bV 

^I^H 

boundary,  or  not.  In  the  former  case  it  is  usual  to  call 
the  whole  continuous,  and  unusual  to  allow  of  its  being 
also  called  discrete ;  but  it  appears  to  me  that  the  notion 
of  continuity  is  derived  from  that  of  contiguity,  and  that 
the  definition  of  contiguity  involves  the  notion  of  a  com- 
mon boundary  between  two  parts ;  hence,  for  the  notion 
of  a  continuous  whole,  I  prefer  to  substitute  the  notion 
of  a  whole  consisting  of  parts,  whose  boundaries  constitute 
a  continuous  Series. 

§  3.  Having  contrasted  continuity  with  discreteness, 
we  will  now  examine  another  meaning  frequently  at- 
tached to  the  word  continuous,  which  may,  I  think,  be 
conveniently  contrasted  with  discontinuous.  The  term 
discrete,  as  hitherto  explained,  applies  to  a  single 
variable  whose  variations  are  not  considered  in  connec- 
tion with  the  variations  of  other  variables.  We  have 
now  to  consider  so-called  correlated  variables,  the 
variation  of  one  of  which  is  dependent  on  that  of  the 
other  according  to  some  assignable  formula.  In  this 
case  we  shall  find  that  while  the  independent  variable 
is  continuous,  the  changes  of  the  other  variable  correlated 
with  this  continuous  series  may  be  either  continuous,  or 
,  (as  it  may  be  described)  discontinuous.  The  dependent 
variable  will  be  said  to  vary  continuously  when  whatever 
section  of  its  actual  variation  is  considered,  every  possible 
value  intermediate  between  those  assumed  at  the  be- 
ginning and  end  of  the  section  are  actually  represented ; 
it  will  be  said  to  vary  discontinuously  when  within  some 
section  of  its  actual  variation  there  are  intermediate 
values  which  are  not  represented.  The  most  familiar 
instance  of  this  kind  of  correlated  discontinuity  or  con- 
tinuity are  those  in  which  Time  is  the  independent 


i66 


CHAPTER  XI 


TEMPORAL  AND  SPATIAL  RELATIONS 


167 


variable.  The  actual  variations  of  Time  are  continuous 
in  the  first  sense  of  the  term,  and  every  possible  value 
is  actually  represented  ;  but  there  may  be  gaps  in  the 
variations  of  the  variable  which  depends,  according  to 
some  formula,  upon  the  variation  of  Time.  A  simple 
illustration  is  that  of  acceleration :  thus  if  a  body  is 
moving  on  a  rough  horizontal  table  until  it  falls  over  the 
edge,  then  at  the  instant  when  it  begins  to  fall  there  is 
discontinuity  in  the  change  of  acceleration.  While  it  is 
moving  horizontally  its  movement  is  subject  to  the 
retardation  of  friction  operating  horizontally  ;  when  it 
is  falling,  on  the  other  hand,  its  movement  is  subject  to 
the  acceleration  of  gravity  which  operates  vertically. 
There  must,  therefore,  be  an  instant  in  which  the 
acceleration  or  retardation  changes  from  one  value  to 
another  with  the  omission  of  all  the  possible  values  in- 
termediate between  the  horizontal  retardation  and  the 
vertical  acceleration.  The  acceleration  in  such  a  case  varies 
discontinuously,  but  not  so  the  velocity,  for  every  possible 
velocity  intermediate  between  the  rate  of  movement  of 
the  body  on  the  table  and  the  rate  when  it  is  beginning 
to  fall,  is  assumed  by  the  body  during  the  intermediate 
time;  for  the  body  does  not  fall  vertically,  but — neg- 
lecting the  resistance  of  the  air — along  a  parabola. 

Now  according  to  a  prevalent  view  in  philosophy,  the 
theory  that  all  change  is  continuous  is  intuitively  axio- 
matic. But  the  change  of  acceleration  in  the  above 
case  would  seem  to  contradict  this  theory ;  although 
physicists  do,  as  a  matter  of  fact,  hold  that  the  change 
of  velocity  is  continuous.  The  explanation  of  the 
apparent  contradictions  to  the  theory  is  to  be  found 
in  the  discontinuity  of  the  physical  processes   corre- 


I  -M 


A 


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A>^ 


U 


\y 


lated  with  the  continuous  series  either  of  Time  or 
of  Space.  For  example,  some  of  the  surfaces  in  Space 
are  boundaries  separating  contiguous  bodies  of  quite 
different  characters,  such  as  solid  and  gaseous;  here 
then,  discontinuity  holds  of  the  dependent  variable 
'physical  character'  as  determined  by  the  independent 
variable  'spatial  position.'  Thus  Time  and  Space  are 
conceived  as  continuous  in  the  first  sense  of  the  term, 
but  the  variations  correlated  with  these  independent 
variables  are  frequently  discontinuous ;  and  when  this 
is  the  case  discontinuity  of  the  variations  correlated  with 
the  variation  of  Time  is  explained  by  the  discontinuity 
of  the  variations  correlated  with  the  variation  of  Space. 
It  should  be  observed  that  when  speaking  of  change 
in  the  surface  of  a  body,  one  part  of  which  is  solid  and 
another  part  liquid,  or  one  part  rough  and  another  part 
smooth,  or  one  part  red  and  another  part  green,  the 
word  change  is  applied  to  variations  correlated  with  varia- 
tion of  Space  instead  of  Time.  We  are  apt  to  regard  the 
words  variation  and  change  as  synonymous,  but  it  is 
very  important  to  restrict  the  term  variation  to  uncorre- 
lated  differences,  and  to  apply  the  term  change  to  dif- 
ferences correlated  either  with  differences  of  Time  or 
with  differences  of  Space.  This  distinction  affords  some 
explanation  of  the  rather  vague  statement  of  philosophers 
that  causal  process  is  continuous :  a  causal  process  is  a 
process  of  change,  and  as  such  is  correlated  with  a 
variation  of  Time  or  of  Space,  Time  and  Space  being 
admittedly  continuous;  but  the  variations  themselves 
in  the  changing  variable  correlated  with  Time  or  Space 
may  be  discontinuous.  To  elucidate  this  seemingly 
paradoxical  notion  of  discontinuous  variation,  let  us 


t 


i68 


CHAPTER  XI 


TEMPORAL  AND  SPATIAL  RELATIONS 


169 


imagine  a  process  of  change  the  beginning  and  end  of 
which  are  dated  by  the  numbers  i  and  3.  If  the  symbol 
a  represents  the  determinate  character  dated  at  i,  and 
the  symbol  f  the  determinate  character  manifested  at 
date  3 — a  and  /"denoting  different  determinates  under 
the  same  determinable,  whose  determinates  a^  b,  c,  d, 
e,  /,,.,  can  be  ranged  in  an  order  depending  upon  a 
comparison  of  degrees  of  difference — then  the  correlated 
discontinuity  is  illustrated  by  assuming  that  during  the 
period  from  i  to  3,  the  stretch  from  ^  to  ^  (containing  d) 
is  not  manifested.  Through  the  period  from  date  i 
to  date  2,  we  will  say,  the  manifestation  changes  con- 
tinuously from  the  character  a  to  the  character  c\  and 
through  the  period  from  date  2  to  date  3,  it  changes 
from  e  to/" also  continuously.  At  no  instant  within  the 
period  from  i  to  3  is  the  character  d  manifested.  At 
the  terminal  phase  belonging  to  the  period  i  to  2, 
the  manifestation  must  be  characterised  as  c\  but  at 
the  initial  phase  belonging  to  the  period  2  to  3,  it  must 
be  characterised  as  e.  As  so  regarded  the  dates  of 
these  two  phases  cannot  be  identified ;  hence  we 
cannot  speak  of  any  determinate  character  as  being 
manifested  at  the  date  2.  On  this  ground  therefore, 
at  the  instant  in  question,  discontinuity  of  variation 
must  be  attributed  to  the  process.  A  similar  illustra- 
tion, with  Space  instead  of  Time  as  the  continuous 
variable,  is  afforded  by  supposing  the  surface  of  a  table 
divided  into  two  contiguous  parts,  one  of  which  is 
entirely  red,  and  the  other  entirely  green,  and  considering 
the  colour  of  the  line  which  is  the  common  boundary 
of  these  two  parts.  This  question  can  perhaps  hardly 
be  said  to  involve  a  paradox,  for  colour  characterises 


0 


^^      ^j 


'i 


i 


i 


i   ^ 


a  surface  and  the  surface-parts  of  a  surface,  it  does 
not  characterise  any  line  which  is  contained  in — but 
is  not  a  part  of — the  surface.  If  this  solution  of  the 
paradox  be  accepted,  by  the  same  method  the  paradox 
involved  in  correlation  with  Time  may  be  removed ;  for, 
just  as  the  parts  of  a  surface  which  are  two-dimensional 
as  is  the  whole  surface,  must  be  distinguished  from 
the  boundary  or  separating  lines  which  are  contained 
in,  but  are  not  parts  of,  the  surface;  so  the  parts  of 
Time  which  are  of  one  dimension  (as  is  the  whole  Time) 
must  be  distinguished  from  the  instants,  which  are  of 
no  dimension,  and  which  are  contained  in  (but  are  not 
parts  of)  Time.  And,  just  as  colour  was  said  to  charac- 
terise the  surface-parts  of  the  table  and  not  the  lines 
contained  in  these  surface-parts,  so  acceleration  must 
be  taken  to  characterise  a  movement  occupying  a 
certain  part  of  Time,  and  not  a  position  of  the  moving 
body  which  is  correlated  with  an  instant ;  for  an  instant 
is  contained  in,  but  is  not  part  of,  the  period  of  Time 
to  which  the  process  of  change  is  referred. 

§  4.  Correlated  continuity  is  probably  always  exhibited 
by  immanent  processes,  i.e.  these  are  non-discontinuous 
in  their  correlation  with  Time.  The  philosophical 
problem  arises  when  transeunt  causality  is  introduced. 
It  may  be  stated  thus:  How  is  it  that  at  a  certain 
moment  of  time,  two  separate  processes  which  have 
been  immanently  determined  previously  to  this  moment 
of  time,  cease  to  be  for  the  subsequent  time  determined 
by  merely  immanent  causality?  This  can  only  be  ex- 
plained by  supposing  some  kind  of  connectional  deter- 
mination; i.e.  if  S  and  T  represent  the  two  immanent 
processes,  in  order  to  account  for  transeunt  action  taking 


170 


CHAPTER  XI 


TEMPORAL  AND  SPATIAL  RELATIONS 


171 


place  at  one  instant  of  time  rather  than  at  another,  we 
must  suppose  some  kind  of  connection  between  the 
manifestations  of  5  and  the  manifestations  of  T\  and 
the  postulate  is  constructed  that  where  there  is  transeunt 
action  of  T  upon  S,  there  is  involved  in  the  formula  of 
determination  a  different  mode  of  connection  from  that 
of  Time.  To  this  form  of  connection  we  give  the  name 
spatial,  because  in  the  case  of  physically  determined 
phenomena  transeunt  causality  always  does  operate  in 
Space.  Defined  relations  of  spatial  connection  enter 
only  in  the  formulae  of  interphysical  causality,  whereas 
every  kind  of  causal  formula  involves  defined  relations 
of  temporal  connection.  In  the  chapter  on  transeunt 
and  immanent  causality  I  have  suggested  that,  for 
interpsychical  causality,  what  takes  the  place  of  spatial 
connection  is  the  attachment  of  both  feeling  and  cogni- 
tion to  the  same  object;  including,  under  the  term 
feeling,  conation  in  its  two  forms  of  attraction  and 
repulsion  felt  with  greater  or  less  intensity  towards  ex- 
periences perceptually  or  imaginatively  apprehended; 
and  under  the  term  cognition,  including  the  variations 
in  degrees  of  determinateness,  as  well  as  of  content, 
according  as  the  experience  is  thought  of  as  having  this 
or  that  character.  Thus  in  my  view  the  formula  of  inter- 
psychical causality,  introducing  variable  relations  of 
feeling  and  cognition  as  causal  determinants,  takes  the 
place  of  variable  spatial  relations  as  causal  determinants 
in  interphysical  causality.  For  simplification  of  this 
exposition  I  have  supposed  the  transeunt  action  to 
operate  ex  abrupto  so  that  the  instant  of  time  at  which 
it  is  dated  can  be  determinately  assigned.  But  our 
account  of  transeunt  action  must  be  extended  to  the 


U  1 


J 


4- 

i 


it 


;< 


17 


t 

4 


^  - 


X 


A,« 


cases  where  some  spatial  relation,  which  either  remains 
unchanged  or  alters  in  the  course  of  time,  continually 
subsists  between  the  manifestations  of  5  and  7",  and 
where  transeunt  action  is  therefore  temporally  con- 
tinuous instead  of  being  instantaneous. 

§  5.    We  may  now  turn  to  the  specific  topic  of  this 
chapter,  and  consider  the  temporal  relation  of  cause  to 
effect  which  is  commonly  said  to  be  that  of  before  to 
after.   In  the  first  place  it  must  be  pointed  out  that  mani- 
festations cannot  be  related  merely  under  the  form  of 
before  to  after,  but  must  always,  in  addition,  be  regarded 
as  manifestations  of  the  same  continuant-entity,  whose 
nature  is  expressed  in  the  formula  according  to  which 
the  preceding  manifestations  determine  the  succeeding. 
And,  in  the  second  place,  parallel  with  the  temporal 
order  amongst  the  manifestations  of  a  continuant,  we 
have  to  consider  the  spatial  order  amongst  the  mani- 
festations of  an  occupant.    In  somewhat  figurative  lan- 
guage we  may  conceive  of  an  occupant  as  manifesting 
itself  in  a  succession  of  regions  which  form  rings  from 
the  more  inner  to  the  more  outer,  separated  byconcentric 
boundaries.    The  relations  of  the  inner  to  the  outer 
manifestations  of  an  occupant  are  analogous  to  the  rela- 
tions of  the  preceding  to  the  succeeding  manifestations 
of  a  continuant.  An  adequate  knowledge  of  the  immanent 
nature  of  the  entity  would  enable  us  to  infer  equally 
from  the  preceding  to  the  succeeding,  as  from    the 
succeeding  to  the  preceding,  in  the  case  of  a  continuant; 
and  from  the  inner  to  the  outer,  as  from  the  outer  to  the 
inner  in  the  case  of  an  occupant.    Thus  reference  to 
changing  and  spreading  characters  to  the  same  con- 
tinuant or  occupant  is  the  basal  principle  underlying 


^ 


172 


CHAPTER  XI 


TEMPORAL  AND  SPATIAL  RELATIONS 


173 


causality.  Now  in  spite  of  this  possibility  of  reciprocal 
inference,  we  nevertheless  regard  the  preceding  as  objec- 
tively determining  the  succeeding,  as  well  as  the  inner  as 
objectively  determining  the  outer;  and  never  reversely, 
the  succeeding  as  objectively  determining  the  preceding, 
and  the  outer  as  objectively  determining  the  inner.  The 
explanation  of  this  refusal  to  reverse  the  order  of  objec- 
tive determination  in  the  temporal  and  spatial  manifesta- 
tions of  continuants  and  occupants  requires  us,  I  think, 
to  pass  from  immanent  to  transeunt  causality.  Thus,  at 
a  certain  moment  of  time,  an  immanent  process  of 
causality  may  be  broken  in  upon  from  without  by  an 
influence  which  modifies  the  succeeding  manifestations, 
so  that  these  are  different  from  what  they  would  have 
been  under  the  uninterrupted  course  of  immanent  process. 
So  while  the  manifestations  which  preceded  the  inter- 
ruption can  be  determined  from  the  mere  knowledge  of 
the  formula  of  immanent  causality,  after  the  interrup- 
tion the  relation  of  the  succeeding  to  the  preceding  is 
objectively  differentiated  from  that  of  the  preceding  to 
the  succeeding.  It  is  to  be  noted  that  the  temporal 
relation  involved  in  transeunt  causality  itself,  is  neither 
that  of  succeeding  nor  of  preceding,  but  of  simultaneity. 
Analogously,  in  the  case  of  the  occupant,  the  reason 
why  we  regard  the  inner  manifestations  as  objectively 
determining  the  outer,  and  not  reversely,  is  explained 
by  the  introduction  of  transeunt  causality.  Just  as,  in 
time,  we  can  take  the  boundary  between  the  preceding 
and  succeeding  phases,  and  show  that  when  a  cause 
from  without  operates  at  this  boundary,  the  succeeding 
phases  are  immediately  modified;  so  we  may  take  the 
surface  of  an  occupant  as  the  boundary  at  which  a 


s 

1 


<i 


cause  from  without  operates,  and  show  that  it  imme- 
diately modifies  the  outer  and  not  the  inner  state  of  the 
occupant.  Simple  illustrations  of  this  principle  are 
afforded  by  an  extensible  string,  or  a  compressible 
fluid:  when  a  string  is  subjected  to  an  equal  tension 
at  both  ends,  the  transeunt  causality  is  from  the  extremes 
to  the  centre,  while  the  immanent  causality,  which  reacts, 
is  from  the  centre  to  the  extremes;  or  again,  when  a 
fluid  is  subjected  to  equal  pressure  throughout  its  surface, 
the  transeunt  causality  is  from  the  outer  surface  to  the 
inner,  while  the  immanent  causality  with  which  the  fluid 
reacts  is  from  the  inner  to  the  outer. 

§  6.    The  above  account  requires  some  explanatory 
modification ;  for  in  all  such  cases  as  those  we  are  con- 
sidering, manifestations  of  an  occupant  which  are  actual 
over  certain  regions  of  space,  at  any  given  time,  are 
potential  over  other  regions  ;  and  similarly,  manifesta- 
tions of  a  continuant  which  are  actual  throughout  certain 
periods    of  time,  are  potential   throughout  the  other 
periods.    This  point  is  illustrated  with  peculiar  signifi- 
cance in  psychology,  where  periods  of  apparent  abeyance 
in  consciousness  of  the  familiar  phenomena  of  associa- 
tion illustrate  in  the  extremest  sense  the  potentiality  for 
manifestations.  The  occasions  when  this  potentiality  is 
converted  into  an  actuality  are  when  an  experience 
breaks  in  upon  the  previous  current  of  thought  and 
operates  transeuntly  in  modifying  the  subsequent  pro- 
cesses.   In  a  precisely  similar  way,  the  occasions  when 
the  potentiality  of  a  body  for  exerting  pressure  or 
sustaining  tension  is  converted  into  actuality,  are  those 
when  it  comes  into  transeunt  contact  with  a  foreign 
body  which  modifies  its  subsequent  states.     In   any 


174 


CHAPTER  XI 


TEMPORAL  AND  SPATIAL  RELATIONS 


175 


case  of  this  kind  we  may  distinguish  those  manifesta- 
tions which  are  modified  by  the  transeunt  action  from 
those  which  could  have  been  determined  without 
knowledge  of  such  action.  It  will  be  found  that  the 
unmodified  manifestations  of  the  continuant  are  related 
to  the  modified  as  earlier  to  later,  and  of  the  occupant  as 
inner  to  outer.  Thus  to  take  the  occupant,  for  example, 
when  a  foreign  body  attracts  a  given  body  as  a  whole, 
it  does  not  affect  the  internal  motions  of  its  parts, 
represented  by  temperature,  chemical  constitution,  inner 
strains  and  stresses,  etc.,  but  only  its  situation  relatively 
to  other  bodies,  and  these  may  be  properly  called  outer 
manifestations  relatively  to  the  inner  and  immanent 
processes  of  the  body.  Contrasting  an  illustration  of 
this  kind  with  such  transeunt  action  as  the  application 
of  heat  to  a  gas,  the  transeunt  causality  in  the  latter  case 
appears  to  produce  effects  in  the  inner  region  as  well  as 
the  outer  occupied  by  the  substance;  but  this  is  because 
the  gas  does  not  in  truth  constitute  a  unit-entity,  and 
must  be  broken  up  into  parts  before  we  can  apply  with 
significance  the  distinction  between  the  immanent  and 
the  transeunt.  From  the  point  of  view  of  mechanical 
and  thermal  analysis  the  parts  into  which  the  gas  must 
be  broken  up  are  molecules  whose  only  inner  and  im- 
manent manifestations  are  chemical.  The  application 
of  heat  affects  the  actions  between  the  molecules  them- 
selves, as  represented  by  their  relative  movements  and 
mutual  pressures,  and  these  illustrate  transeunt  causality, 
the  chemical  or  inner  processes  of  the  molecules  being 
left  unaffected  to  follow  their  own  immanent  course. 
The  case  of  the  gas,  then,  when  properly  analysed,  is  a 
further  illustration  of  the  principle  that  the  transeunt 


^) 


I 


i' 


\ 


rt    A 


1 


processes  modify  the  outer  and  the  later  manifestations 
without  affecting  the  inner  and  the  earlier;  and  that 
between  the  transeunt  cause  and  the  transeunt  effect 
there  is  temporal  simultaneity  and  spatial  coincidence. 
§  7.  The  above  illustration  of  transeunt  and  immanent 
action  suggests  a  third  kind  of  causality  which  requires 
separate  consideration,  viz.  that  involved  in  the  com- 
pression of  a  compressible  solid,  as  distinguished  from 
the  compression  of  a  gas.    Here  the  correlated  con- 
ceptions of  stress  and  strain  are  properly  applied;  a 
liquid  or  a  solid  when  it  is   unnaturally  compressed, 
exerts  a  force  of  expansion  which  decreases  from  a 
certain  maximum  to  the  minimum  zero,  as  the  com- 
pression is  allowed  to  decrease.  And  a  solid  or  extensible 
string,  when  it  is  unnaturally  extended  exerts  a  force  of 
contraction  which  decreases  from  a  certain  maximum 
limit  to  the  minimum  zero  as  the  extension  is  allowed  to 
decrease.     It  will  help  us  to  understand  the  nature  of 
the  force  of  tension  illustrated  by  the  string  if  we  contrast 
it  with  the  force  of  attraction  ;  for,  while  between  two 
attracting  bodies  the  force  of  approach  is  stronger  the 
nearer  they  are  to  one  another,  between  two  parts,  say,  of 
a  string  the  force  of  approach  is  stronger  the  further  they 
have  been  pulled  from  one  another.    Now  to  understand 
the  type  of  causality  operating  in  the  case  of  the  com- 
pressed solid,  we  may  mentally  divide  the  volume  which 
it  occupies  into  an  inner  core  and  an  outer  ring.    The 
effect  of  the  pressure  operating  from  the  foreign  force 
is  to  unnaturally  contract  the  volume  occupied  by  the 
inner  core,  causing  an  outward  pressure  upon  the  inner 
side  of  the  outer  ring.  Apart  from  the  outward  pressure, 
the  effect  of  the  inward  pressure  would  be  manifested 


I 


176 


CHAPTER  XI 


TEMPORAL  AND  SPATIAL  RELATIONS 


177 


in  the  restoration  of  the  outer  ring  to  its  natural  size ; 
we  may  therefore  properly  take  the  contracted  condition 
of  the  inner  core  as  the  immanent  cause  of  the  subsequent 
expansion  of  the  outer  ring.  This  form  of  causality 
illustrates  the  same  type  of  analogy  between  temporal 
and  spatial  factors  as  we  have  already  noted,  the  inner 
in  space  corresponding  to  the  earlier  in  time,  and  the 
outer  in  space  corresponding  to  the  later  in  time,  while 
the  causality  operating  at  the  common  boundary  between 
the  inner  core  and  the  outer  ring  corresponds  to  the 
moment  of  time  at  which  the  condition  of  the  inner 
core  influences  the  condition  of  the  outer  ring.  If  then 
immanent  causality  alone  were  involved,  our  knowledge 
of  the  shape  and  size  of  the  inner  core  would  determine 
for  us  a  knowledge  of  the  subsequent  and  contiguous 
shape  and  size  assumed  by  the  outer  ring.  But  when 
the  transeunt  causality  from  the  foreign  force  is  brought 
into  consideration,  the  subsequent  and  contiguous  shape 
and  size  assumed  by  the  outer  ring  is  modified.  Having 
divided  at  an  arbitrary  surface  the  inner  core  from  the 
outer  ring,  we  must  make  a  correspondingly  arbitrary 
division  in  time  between  the  earlier  and  the  later  states 
of  the  body.  Considering  the  solid  body  alone,  the 
inner  core  is  first  under  a  pressure  dependent  upon  its 
unnatural  shape  and  size,  and  the  subjection  of  the  outer 
ring  to  the  foreign  compressing  force  of  pressure  is 
later  in  time.  So  the  inner  surface  of  the  outer  ring  at 
the  earlier  stage  is  pressing  outwards,  and  the  outer 
surface  of  this  outer  ring  at  the  later  stage  is  pressing 
inwards.  Hence  the  pressure  at  the  inner  surface  at  the 
earlier  stage  represents  that  part  of  the  process  (deter- 
mined solely  by  immanent  conditions)  which  is  unmodi- 


•) 


V 


f 

f 

■4 


4 


} 


:i     > 


If 


fied  by  transeunt  action ;  while  the  pressure  at  the  outer 
surface  at  the  later  stage  represents  that  part  of  the 
process  which  is  modified  by  transeunt  action.  The 
case  of  the  extensible  string  is  capable  of  precisely 
similar  analysis ;  so  also  is  that  of  the  varying  tempera- 
ture of  gas  enclosed  in  an  envelope.  In  all  these  cases, 
the  immanent  tendencies  operate  in  the  direction  of  an 
assignable  form  of  equilibrium,  and  by  dividing  the 
whole  process  into  temporal  and  spatial  parts  corre- 
sponding to  one  another  we  shall  always  find,  by  taking 
the  earlier  stage  to  correspond  with  the  inner  region, 
and  the  later  stage  with  the  outer  region,  that  the  former 
represents  the  part  of  the  process  unmodified,  and 
the  latter  the  part  of  the  process  modified  by  transeunt 
action. 

A  failure  in  the  analogy  between  Space  and  Time 
hitherto  unnoted,  may  here  be  pointed  out.  Whereas 
the  dating  of  a  process  is  absolute,  in  the  sense  that  it 
is  independent  of  the  continuant  to  which  the  process 
refers,  the  locating  of  a  process  as  being  relatively  inner 
or  outer  is  not  absolute,  for  what  is  relatively  inner  to  one 
occupant  is  relatively  outer  to  another.  To  establish 
the  required  analogy,  it  would  be  necessary  to  conceive 
that,  of  two  temporally  distinguished  parts  of  a  process 
extending  through  Time,  that  which  is  earlier  when 
referred  to  one  continuant  is  later  when  referred  to 
another ;  just  as,  of  two  spatially  distinguished  parts  of 
a  process  extending  through  Space,  that  which  is  inner 
when  referred  to  one  occupant  is  outer  when  referred 
to  another. 


J  L  III 


12 


APPENDIX  ON  EDUCTION 

§  I.  In  the  problem  before  us  we  shall  be  concerned 
with  a  certain  adjectival  determinable  P  which  has  a 
determinate  values— A,  A»  ..-A— ^nd  shall  proceed  to 
consider  J/ instances,  each  of  which  is  characterised  by 
one  or  other  of  these  a  determinate  characters. 

Any  actual  set  of  occurrences  of  length  M  will  exhibit 
a  certain  proportion  among  the  a  determinate  charac- 
ters;—^! occurrences  of  A,  m^  of  A  ...  m^  of  A  (say), 
where 

The  proportion  m,\m,\...\  m,  exhibited  in  M  occur- 
rences will  be  denoted  by  \l. 

These  occurrences  also  will  be  presented  in  a  definite 

The  order  in  which  the  occurrences  exhibitmg  the 

proportion  ^  are  presented  will  be  denoted  by  /x. 

The  following  two  elementary  arithmetical  formulae 
will  be  required: 

( I )    Combination-formula . 

The  number  of  integral  solutions  of  the  equation 

i.e.  the  number  of  values  that  /x  may  assume  in  M 
occurrences,  is 

a(a-hi)...(a  +  i^-i) 


^< 


<\\ 


y^ 


4 


J 


APPENDIX  ON  EDUCTION 
(2)   Permutation-formula, 


179 


The  number  of  different  orders /x,  in  which  a  given  pro- 
portion m^\m^:  ,.,  :m^  may  be  presented  in  M  occur- 
rences, is 

M\ 
m^\  m^\ ...  m^\  * 

§  2.  Probability  is  a  magnitude  to  be  attached  to 
any  possibly  true  or  possibly  false  proposition ;  not,  how- 
ever, to  the  proposition  in  and  for  itself,  but  in  reference 
to  another  proposition  the  truth  of  which  is  supposed 
to  be  known.  For  example,  the  probability  of  the  pro- 
position that  *The  next  throw  of  a  certain  coin  will 
yield  head'  may  have  its  value  assigned  by  the  know- 
ledge that  'It  will  yield  either  head  or  tail.'  The  value 
of  the  probability  as  so  determined  is  not  necessarily 
the  same  as  that  determined  by  the  knowledge  that 
*The  previous  throws  of  the  coin  have  presented  heads 
and  tails  with  a  certain  frequency.'  The  proposition 
to  which  the  probability  attaches  will  be  conveniently 
termed  the  proposal]  and  the  proposition  to  which  the 
probability  refers  as  that  whose  truth  is  supposed  to  be 
known  will  be  conveniently  termed  the  supposaL 

Furthermore,  in  order  that  the  notion  of  probability 
shall  have  significance,  it  is  requisite  that  the  proposition 
standing  as  supposal  shall  not  be  known  to  be  false.  Using 
the  notation  adopted  by  Mr  J.  M.  Keynes,  which  in- 
troduces the  solidus : 

pjs  symbolises  the  probability  of  the  proposal  /  as 
depending  upon  or  referred  to  the  supposal  s. 

The  notation//^"  may  be  read  */  upon  s'  or  p  given  s' 


12—2 


I 


i8o 


APPENDIX  ON  EDUCTION 


APPENDIX  ON  EDUCTION 


i8i 


^  I 


The  maximum  limiting  magnitude  oipjs  is  certitude : 
viz.  when  the  truth  of/  is  implied  by  s.  Its  minimum 
limiting  magnitude  is  contra-certitude :  viz.  when  the 
falsity  of/  is  implied  by  s.  Since  probability-values  are 
signless,  the  minimum  value  (contra-certitude)  must 
always  be  represented  as  zero ;  and,  since  certitude  is  the 
maximum  probability-value,  all  other  probability-values 
are  (proper)  fractions  of  certitude.  It  is,  in  fact,  usual 
to  express  probability-values  as  pure  fractions,  such  as 
:|  or  I ;  and  to  express  certitude  by  unity.  But  this  repre- 
sentation is  logically  false,  and  should  only  be  permitted 
as  a  convenient  abbreviation. 

In  estimating  the  probability  of/  as  depending  on 
the  specific  knowledge  s  it  is  essential  that  s  should 
represent  the  whole  of  the  supposed  knowledge,  rele- 
vant to  the  case.  Briefly,  the  dependence  indicated  by 
the  equation  p\s  =  f  of  certitude  (say),  when  expressed 
as  an  implication,  means : 

\{  s  alone  were  known,  then  the  probability  of/  would 
be  §  of  certitude. 

If  /  also  were  known,  p\st  would  not  necessarily  be 

the  same. 

In  this  respect,  the  relation  of  dependence  for  prob- 
ability is  to  be  contrasted  with  the  relation  of  impli- 
cation.   Thus 

*/  is  implied  by  s'  corresponds  to //^  =  certitude. 

Now,  if  */  is  implied  by  s^  then  also  '/  is  implied 
by  si'  and  hence //^^  also  =  certitude. 

In  other  words,  additional  supposed  knowledge  can- 
not alter  the  degree  of  probability  of  any  proposition 
known  to  be  true  or  to  be  false,  but  it  may  always  alter 
the  degree  of  probability  of  a  proposition  not  known  to 
be  true  or  to  be  false. 


i<^ 


«i 


t., 


i 


i\ 


i 


\\ 


.{', 


\ 


/^ 


\  3.  Two  axioms  are  required  for  the  working  of  the 
probability-calculus :  viz.  the  additive  and  the  multipli- 
cative. With  the  notation  above  explained,  these  axioms 
may  be  formulated  as  follows : 

Additive  axiom: 

If/  and  q  are  known  to  be  not  both  true,  then 

Multiplicative  axiom: 

lip  is  not  known  to  be  false,  then  ^ ^ 

(/  and  q)\h  ^pjk  x  ql{p  and  h). 

When  such  symbols  as  /,  q  stand  for  propositions, 
the  conjunctive  */  and  q '  will  be  abbreviated  into  pq. 
But,  when  x,  y  (say)  stand  for  quantities.  then;i; .  y  or  xy 
will  mean  'xy^y!  On  the  other  hand,  'pox  q'  should 
never  be  written  'p^q'\   nor  should  pjq  be  written 

p^q  or^  (in  spite  of  certain  analogies). 

Thus  the  formula  for  multiplication  may  be  written 

pqjh  =p\h  X  q\ph, 

§  4.    The  following  corollaries  will  be  required  in  the 

sequel : 

CoR.  I.    If  A  or/2  or  .../r^/,  where /i,  A,  .../r  are 
co-disjunct,  then,  by  additive  axiom, 

plh  =pjk  ^pjk  -h  . . .  +am. 

Cor.  2.    If,  further, 

p,lh^p,\h^...^pr\K 

then  each  of  these  ^^'-^  . 

CoR.  3.    If  V  implies/,'  \,e,q  =  p  and  q,  then 
qjk  =  (/  and  q)lh  =plh  x  qjph, 
by  multiplicative  axiom. 


i82  APPENDIX  ON  EDUCTION 

Cor.  4.    If  s^  or  ^2  or  ...  ^r=  Sy  where  s^y  s^,  ...  s^  are 
co-disjunct,  and  if,  further, 

then  each  of  these  —pjsL 

For,  let  each  of  the  above  =  x.    Then 

/V'^  =ps,lk  ^psjk  + . . .  +/^,/>^, 
i.e.    sjh .  //^>4  =  ^y^ .  pjs.h  +  V'^  .//^^^  +  . . .  +  ^,/>4  .//^,>4 

=  (^j/^  +  sJh  +  . . .  +  i-^^ )  ;i; 
=  sjk .  ^. 

.-.    X^plsk  Q.E.D. 

§  5.  Now  the  axioms  of  probabiHty  enable  us  to 
infer  any  probability-conclusion  only  from  probability- 
premisses.  In  other  words,  the  calculus  of  probability 
does  not  enable  us  to  infer  any  probability- value  unless 
we  have  some  probabilities  or  probability  relations^W«. 
Such  data  cannot  be  supplied  by  the  mathematician. 
E.g.  the  rules  of  arithmetic  and  the  axioms  of  the  prob- 
ability-calculus are  utterly  impotent  to  determine,  on 
the  supposed  knowledge  that  the  throw  of  a  coin  must 
yield  either  head  or  tail  and  cannot  yield  both,  the 
probability  that  it  will  yield  head  or  that  it  will  yield 
tail.  We  must  assume  that  the  two  co-exclusive  and 
co-exhaustive  possibilities  are  equally  probable,  before 
we  can  estimate  the  probability  of  either  as  being  a  half 
of  certitude.  The  assumptions  ultimately  required  must 
be  regarded  as  PostulateSy  and  their  critical  examination 
will  not  here  be  entered  upon.  The  working  postulates 
are  such  as  assert  equiprobability  amongst  alternative 
possibilities;  and  constitute  what  may  be  called,  in 
Mr  Keynes's   terminology,  postulates  of  indifference. 


APPENDIX  ON  EDUCTION 


183 


v\ 


f-i' 


The  precise  form  of  the  postulate  required  in  each  par- 
ticular application  must  be  justified  by  the  special  nature 
of  the  case.  We  shall  immediately  lay  down  the  two  pos- 
tulates employed  in  the  theory  of  eduction,  postponing 
for  the  present  any  further  philosophical  discussion. 

§  6.  The  two  following  postulates  in  the  Theory  of 
Eduction  are  concerned  with  the  possible  occurrences 
of  the  determinates/,,/,,  ..-A  under  the  determinable 
P,    The  symbols  of  §  i  are  employed. 

( 1 )  Combination-Postulate. 

In  a  total  of  M  instances,  any  proportion,  say 
m^\m^:.,.  :w., where  m^  +  m^-\- ..,-\-ni,  =  My  is  as  likely 
as  any  other,  prior  to  any  knowledge  of  the  occurrences 
in  question. 

(2)  Permutation- Postulate, 

Each  of  the  different  orders  in  which  a  given  pro- 
portion m^\m^:...\mJoY  M  instances  may  be  presented 
is  as  likely  as  any  other,  whatever  may  have  been  the 
previously  known  orders. 

In  what  follows  certitude  will  be  represented  by  unity. 

By  (i),  The  probability  of  any  one  proportion  in  M 
instances 

.       •    M\ 

""a(a+l)(a+2)...(a-hi^-i)' 

By  (2),  The  probability  of  any  one  permutation  in 
which  the  proportion  m,\m.:...r.m,  in  M  instances  may 
be  presented 

m-^\m^\m^\  ..,mj. 
M\ 


i84 


APPENDIX  ON  EDUCTION 


§  7.   Adopting  the  notation  explained  above,  these 
postulates  may  be  symbolised: 

(i)  uM  =  ^ ^' 

^  ^'        a(a+l)...(a  +  J/-i)' 


(2)  ix/ixk  = 

Now  (see  Cor.  3) 


M\ 


^-"^  '^'"'     a(a+i)...(a  +  J/-i)- 

Formula  (3)  gives  the  prior  probability  that,  in  a  set 
of  M  instances,  the  characters/,,/,,  .. ./^  under  P shall 
occur  in  a  determinately  assigned  sequence  in  which  the 
proportion  and  the  permutation  of  these  character- 
manifestations  are  both  fixed. 

Taking   N  instances    (next   following   the    M  in- 
stances) presenting  the  proportion  n,:n,: .,.  :n,,  where 
«i  -f  ;^2  -h  ...+«.  =  A^,  the  principle  of  formula  (3)  may  be 
extended  to  M-\-  N  instances. 
Thus 


(4)     IX  +  v/k 
Now 


a{a+l),.,{a  +  M-^N-i)      ' 


fi  +  v/A  =  {fi  and  p)/A  =  fx/A  x  v/fxA,     by  Mult,  axiom. 
Hence  (5) ^/^  =  '^-^^  =  fe±^' . . .  (^«  +  «») ' 


IJi/A 


a  • 


X 


by  (3)  and  (4). 

From  formula  (5)  which  gives  v/^k  we  proceed  to 
find  the  value  of  v/fi&  which  will  complete  our  solution. 


4ri 


i'l 


If.Vl 


l*M 


»< 


APPENDIX  ON  EDUCTION 


185 


Let  Va,  ^6,  I'c,  etc.  represent  the  different  possible 
permutations  of  the  same  proportion  v=^n^\n^\  ,.,\n^. 

The  number  of  these  is  — \ — ~ ; . 

^i !  «2  •  •  •  •  ^« ' 

Now  expression   (5)  is  independent  of  the  orders 
Vay  ^bt  Gtc.    Hence 


Hence,  by  Cor.  2, 

(6)     v/fiA  =  (i/„  or  z/ft  or  i/^  or  etc.)///.^ 


n^ln^l  ,..n^\ 


N\ 


(J/+a)(J/-f  a-h  l)  ...  {M+N+a-  l) 

(;;^i  +  ;^i) !      {m^  +  t^  J  ! 


I«  I    ••• 


/«i  1  /^i !  m^\n^\ 


l«  I    • 


Again,  let  /i^,  ^2,,  /i^*  etc.    represent   the   different 
possible  permutations  of  the  proportion 

Now  expression  (6)  is   independent  of  the  orders 
Ma,  Mi,,  etc.    Hence 

v\^^h  =  vl^L^h  =  vj^L^h  =  etc. 
Hence,  by  Cor.  4, 

(7)     ^/M  =  ^/(Ma  or  /Aft  or  /LL,  or  etc.)  h 

=  ^/Ma>^ 

^ m 

{M+a)(M+a-h  l)  ...(J/-f-iV+a-l) 

X  (^1  +  ^1)  •       (^g  +  ^g)  ! 


i86 


APPENDIX  ON  EDUCTION 


APPENDIX  ON  EDUCTION 


187 


This  provides  the  required  formula,  viz. : — 
The  probability  of  any  proposed  proportion  in  N 
unexamined  cases  as  depending  upon  any  supposed  pro- 
portion in  M  examined  cases. 

It  will  be  observed  that  the  highest  value  of  this 
probability,  \{  N=M,  is  given  by 

i.e.  the  most  likely  proportion  for  the  new  cases  is  the 
proportion  holding  of  the  known  cases. 

And,  generally,  the  more  closely  v  agrees  with  /x,  the 
greater  is  the  probability  that  v  will  be  true  when  /x.  is 
known  to  be  true. 

§  8.    Elucidation  of  the  formula  for  vj^ih. 

As  above,  we  see  that  vjix^h  =  vjixk. 
Taking  Nxo  be  successively  i,  2,  3,  etc.  the  simplicity 
of  the  above  results  will  be  readily  seen. 

Thus,  for  A^=  I,  the  different  values  of  the  proposal 

1/ are /i, /a,  .../..    Thus 

For  N=-  2,  the  different  values  are  the  dual  permu- 
tations p^p^]  p^p^\  '*'pipa\  AA;  AAJ  '*'P^Pa^  etc.,  etc. 
Thus 

^'P'l^^  "  {M+a){M+a+i)  ' 


p^p^llih 


_     {m^-\-i){m^-i'  i) 
""(J/+a)(J/+a+l)^ 


X      .     /       7  (^2+    l)(^2+2) 


♦\ 


41 


For  iV=3,  the  probabilities  of  the  triple  permuta- 
tions are: 


AAA//^^  = 


A  A  A//^^  = 


{M+a){M-\-a-^l){M+a-\-2)' 


( J/-I-  a)  ( J/-I-  a  +  I )  (i^/+  a  +  2)  ' 


etc.. 


etc. 


etc.. 


etc., 


etc. 


if 


By  addition  of  the  values  for  iV=  2,  we  obtain  those 
for  iV=  I.  And,  by  addition  of  the  values  for  7^=  3,  we 
obtain  those  for  W=2.  And  so  on.  In  this  way  the 
correctness  of  each  formula  is  verified. 

Moreover,  all  specific  results  of  the  formula  giving 

v/fiA  may  be  schematised — if  we  typify  occurrences  as 
draws  from  a  bag  containing  an  indefinite  number  of 
balls  of  the  different  colours  A  iA»  •••A — ^Y  supposing 
a  model  bag  containing  at  first  a  balls  of  different  colours. 
As  each  new  ball  is  drawn  from  the  real  bag,  its  colour 
is  observed  and  it  is  transferred  to  the  model  hdig.  Then, 
the  probability  of  any  proposed  colour  being  drawn  from 
the  real  bag  is  the  same  as  that  of  its  being  drawn  from 
the  model  bag. 

§  9.  The  type  of  case  for  which  the  two  Postulates 
are  permissible  may  be  thus  described. 

It  is  known  that  there  are  certain  conditions  which 
are  constant  in  all  the  occurrences  that  may  take  place 
and  to  which  our  observations  and  inferences  refer.  It 
is  also  presumed  that  these  permanent  or  constant  con- 
ditions are  such  as  tend  to  produce  a  certain  (but  un- 
known) proportion  amongst  the  manifested  characters 
within  any  given  set  of  M  occurrences.  On  the  other 
hand,  each  individual  occurrence  is  actually  occasioned 
by  variable  conditions,  which  are  causally  independent 


i88 


APPENDIX  ON  EDUCTION 


APPENDIX  ON  EDUCTION 


189 


of  one  another,  and  are  such  that  no  prediction  as  to 
their  result  in  any  one  case  can  be  made. 

In  such  a  typical  state  of  affairs,  what  is  unknown  is 
the  proportion  which  tends  to  be  exhibited  owing  to 
the  unchanged  or  permanent  set  of  causal  conditions. 
While,  therefore,  the  determinate  issue  in  any  set  of 
instances  is  causally  independent  of  what  has  previously 
occurred,  yet  it  is  epistemically  dependent;  i.e.  from  the 
point  of  view  of  knowledge,  the  observation  of  previously 
examined  instances  rationally  influences  our  estimate  of 
probability  in  regard  to  what  will  subsequently  occur. 

The  first  postulate,  that  (in  M  cases)  one  proportion 
is  as  likely  as  another,  is  negatively  justified  by  our 
ignorance  of  the  proportion  which  the  permanent  con- 
ditions tend  to  produce.  And  our  second  postulate,  that 
one  order  in  which  any  proportion  may  be  manifested 
is  as  likely  as  any  other,  is  positively  justified  by  our 
knowledge  that  the  variable  conditions  which  occasion 
each  individual  occurrence  are  ontologically  independent 
of  those  which  occasion  any  other  individual  occurrence. 

When  it  is  said  that  the  permanent  conditions  tend 
to  produce  a  certain  proportion /^  1/2  •  •••  "A*  ^X  ^^^  ^^ 
is  not  meant  that  such  a  proportion  will  be  more  nearly 
approached  as  the  series  is  indefinitely  prolonged.  For, 
on  the  contrary,  in  2 J/ cases  the  proportion /^  i/g : ...  \p^ 
is  very  much  less  likely  to  be  exhibited  than  in  incases; 
since  the  number  of  arithmetically  possible  proportions 
is  much  greater  in  a  total  of  2M  than  in  a  total  of  M. 

We  may  finally  point  out  that  the  type  of  case  for 
which  our  theory  of  eduction  holds  may  be  figuratively 
represented  by  imagining  a  die,  in  the  form  of  a  solid 
polyhedron,  whose  plane  faces  are  not  more  than  a  in 
number.    Moreover,  the  die  is  not  known  to  be  either 


f\ 


^vV, 


physically  or  geometrically  regular.  Each  throw  of  the 
die  represents  an  occurrence ;  and,  according  as  the  die 
falls  upon  one  or  another  plane  face,  we  represent  the 
occurrence  as  being  characterised  by  one  or  another  of 
the  a  possible  determinate  adjectives— ^,, /a,  .../a- 

The  constancy  of  the  physical  and  geometrical  pro- 
perties of  the  die  corresponds  to  the  constancy  of  those 
unchanged  causal  conditions  upon  which  the  occurrences 
depend ;  while  the  variable  and  unassignable  impetus  of 
each  toss  corresponds  to  the  variable  condition  which 
occasions  the  actual  issue  in  each  individual  occur- 
rence:— the  varying  condition  which  determines  the 
issue  in  any  one  case  being  causally  independent  of 
that  which  determines  the  issue  in  any  other  case. 


INDEX 


191 


INDEX 


Actual  and  potential  manifestations 

87 
Adjective,  discrimination  of  53 

Agent  and  patient  xxiii,  69,  86 

Agreement,  use  in  induction  24,  28; 

modification  of  figure   61 ;    and 

variancy  37,  48,  51 
Analogy,  relation  to  induction  46 
Analysis  of  instances  24 
Arithmetic  and  logic  xiv 
Article,  see  Predesignations 
Attention,  its  effect  and  cause  in, 

158;  and  freewill  123 

Bacon's  *Table  of  Degrees'  21 
Body  and  mind  xxvi,  151,  159 

Cause,  agents  69,  86 ;  Aristotelean 
classification  73;  and  conation 
120,  150;  convergence  of  factors 
143;  efficient  139;  and  events 
69;  and  foreknowledge  109,118; 
and  implication  4,  144;  parallel 
processes  144;  and  properties  7 1 ; 
temporal  paradox  of  70 

Causes  and  effects,  complete  assign- 
ment of  56, 63;  'different'  defined 
59;  distinction  between  79,  132; 
homogeneity  69;  plurality  of  54; 
reciprocity  of  56,  62  ;  spatio-tem- 
poral relations  between  79,  132 

Change,  and  cause  66,  76,  85 ;  a 
complex  notion  156;  continuity 
of  162 ;  and  the  determinable  84; 
and  continued  existence  68,  80, 
100;  discontinuity  in  166;  modern 
views  on  100 

Characterisation  and  inherence  66 


Cognition,  and  conative  deliberation 
124;  and  final  cause  139,  159; 
in  interpsychical  causality  170; 
and  neural  process  105,  113 

Complementary  universals  62 

Conation,  a  form  of  causality  118 ; 
and  judgments  of  value  126 

Concrete  and  abstract  i 

Contingency,  nomic  10 

Continuant,  and  causal  agency  86 ; 
and  change  81 ;  concept  criticised 
10 1 ;  defined  98;  and  the  deter- 
minable 85  ;  identity  of  79,  128  ; 
and  immanent  causality  97,  128; 
and  law  82;  relation  to  occur- 
rent  xviii,  66,  95 ;  physical  illus- 
tration of  78 ;  psychical  illustra- 
tion of  82 ;  psychical  and  physical 
contrasted  xxii,  137;  and  sub- 
stance 80 ;  relation  to  substantive 
67 ;  and  systems  of  sub-continu- 
ants 92 

Continuity  and  discreteness  162; 
and  discontinuity  165,  167 

Convergent  cause  process  143 ;  il- 
lustrated 147 

Conversion,  fallacy  of  54 

Determinable,  and  change  84 ;  and 
the  continuant  xix,  67,  85 ;  and 
the  given  3 

Determinism  and  freewill  xxxiii, 
123 

Difference,  and  independence  38; 
use  in  induction  25,  29 

Discrimination  53 

Divergent  effect  process  143;  psy- 
cho-physical illustration  149 


'♦'I 


Dualism  xix ;  objections  to  xxxi 

Eduction,  defined  43;  and  induc- 
tion 47 

Effects,  divergence  of  143;  parallel 
processes  144;  plurality  of  54; 
and  properties  72 

Effort,  function  of  107 ;  types  of  1 12 

Ego,  as  agent  103;  as  continuant 
136 

Emotion,  causal  relation  of  factors 

135,  152 
Events  and  occurrents  xxi;  psychi- 
cal and  physical  xxii;  relations 
between  69,  143 
Evidence,  complementary  52 
Evolution,  and  absolute  time  22 
Existents  and  co-existence  71 
Experiment  and  natural  law    18; 

and  transeunt  causality  129 
Extensional  aspects  of  eduction  46; 
wholes  10 1 

Facts,  the  test  of  formulae  33 ;  re- 
lations between  161 

Factual  determination  161 ;  propo- 
sitions I,  3;  universals  6,  11 

Force,  actuality  of  87 ;  stress  and 
strain  175 

Freewill  xxxiii,  123 

Functions  and  properties  97 

'Given'  and  *real'  3 

Hume,  on  causality  4 
Hypothesis,  meanings  of  30 

Identity  and  otherness  xv 

Immanent  causality,  and  the  con- 
tinuant 94,  97,  128;  in  physics 
129;  in  psychology  102,  136; 
function  in  science  140;  and  sys- 
tems of  sub-continuants  92,  141; 
and  time  177;  and  transeunt 
xxiv 


Implication  and  cause  4 

Independency  and  difference  38, 48, 
51,  65;  orders  of  41 

Induction,  and  analogy  46 ;  criteria 
of  19;  and  eduction  47;  use  of 
experiment  in  17,  39 ;  hypotheti- 
cal 30 

Inference  and  divergent  causality 

151 
Instantial  premiss  44 

Intensional  aspects  of  eduction  46 

Intermediary  premiss  46,  50 

Invariability  of  sequence  xxxii 

James,  W.,  on  conative  conflict  122; 

on  emotion  134 
Jevons  on  induction  31 

Kant  on  causality  4 ;  on  substance 
98 

Language,  relation  to  thought  116 

Law,  and  experiment  17;  and  fac- 
tual universals  1 1 ;  logical  form 
of  4,  8 ;  and  uniformities  of  co- 
existence 70 

Logic  as  philosophy  xvi 

Logical  determination  161 

Mathematics  and  logic  xv 

Matter,  and  mind  151,  159;  pri- 
mary qualities  of  88;  spatial  forms 
of  91 

Mill  on  causality  5,  70;  on  freewill 
xxxiv;  on  syllogism  43 

Movement,  and  convergent  causal- 
ity 146,  153;  logical  account  of 
78,  127 

Nomic  necessity  and  contingency 

9 
Number  xv ;  and  order  163 

Objective  determination  4,  69,  172 
Occupant  95,  171 


192 


INDEX 


Occurrent,  and  continuant  66,  78, 
95;  and  event  xxi;  and  transience 
68 

*  Of,'  meaning  of  xx 

Order,  serial  162 

Parallelism,  psycho-physical  xxviii, 
105,  112;  diagram  for  143,  156; 
illustrated  118,  134;  and  infer- 
ential determination  146,  148 

Particulars,  inference  from  44 

Parts  and  boundaries  163,  169 

Permanence  and  change  68,  100; 
of  substance  99 

Plurality  of  Causes,  defined  54; 
illustrated  60 

Possible,  meanings  of  6;  and  po- 
tential 14;  range  of  13 

Postulate  defined  xviii 

Potentiality  defined  14;  and  second- 
ary qualities  88;  and  transeunt 
causality  174 

Predesignations  and  reference  3 

Probability  and  eduction  48;  and 
evidential  data  49 

Problematic  induction,  principle  of 

34 

Properties  and  cause  71,  86;  rela- 
tion to  continuant  68,  96 ;  formula 
for  86,  97 ;  denoting  potentialities 
88,  138 

Psychology,  immanent  process  in 

102,  139;    transeunt  process   in 

103,  173 
Purpose  xxviii 

Separation  53 ;  in  fact  and  thought 

164 
Series,  discrete  and  continuous  162 


Space,  order  in  163 
Specification,  principle  of  21 
Spinoza  xix 
Subjective  activity,  and  cause  103, 

109 ;  and  effort  107 ;  types  of  1 1 1 
Substance,   Kant's  views    on    98 ; 

metaphysical  notion  of  80 
Substantival  identity  79;  separation 

53 

*  Thing'  or  continuant  98 

Thought,  see  Cognition 

Time  and  causality  xxv,  70,  74,  171; 
and  the  continuant  67 ;  order  in 
164;  an  independent  variable  166 

Time  and  Space,  conditions  of 
otherness  79;  forms  of  nexus  90; 
not  analogous  177;  and  transeunt 
causality  172 ;  and  correlated 
variables  167 

Transeunt  causality  xxiii;  relation 
to  immanent  xxiv;  illustrated 
from  physics  130,  174;  in  psy- 
chology 102,  136,  173;  and  psy- 
cho-physical process  104;  func- 
tion in  science  141 ;  and  systems 
of  sub-continuants  92,  141 ;  and 
temporal  relations  172 

Uniformities,  causal  94 ;  of  co-ex- 
istence 70,  74,  94 

Value,  judgments  of  125 
Variables,  discontinuous  165 
Variancy  and  agreement  37,  48,  51 
Verification  31 

Wholes,  extensive  and  extensional 
164 


CAMBRIDGE*.   PRINTED  BY  W.  LEWIS  AT  THE  UNIVERSITY  PRESS 


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DATE  DUE 


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JUl  5     t95f 


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